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Mathematics LibreTexts

1.E: Functions (Exercises)

 

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1.0: Prelude to Functions

In this chapter, we will explore functions that are a kind of relationship between parameters and their properties.

 

1.1: Functions and Function Notation

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Section Exercises

Verbal

Exercise 1.1.1

What is the difference between a relation and a function?

Solution

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

Exercise 1.1.2

What is the difference between the input and the output of a function?

Exercise 1.1.3

Why does the vertical line test tell us whether the graph of a relation represents a function?

Solution

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

Exercise 1.1.4

How can you determine if a relation is a one-to-one function?

Exercise 1.1.5

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Solution

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

 

Algebraic

For the following exercises, determine whether the relation represents a function.

Exercise 1.1.6

\(\{(a,b), (c,d), (a,c)\}\)

Exercise 1.1.7

\(\{(a,b),(b,c),(c,c)\}\)

Solution

function

For the following exercises, determine whether the relation represents y as a function of x.

Exercise 1.1.8

\(5x+2y=10\)

Exercise 1.1.9

\(y=x^2\)

Solution

function

Exercise 1.1.10

\(x=y^2\)

Exercise 1.1.11

\(3x^2+y=14\)

Solution

function

Exercise 1.1.12

\(2x+y^2=6\)

Exercise 1.1.13

\(y=−2x^2+40x\)

Solution

functio

Exercise 1.1.14

\(y=\dfrac{1}{x}\)

Exercise 1.1.15

\(x=\dfrac{3y+5}{7y−1}\)

Solution

function

Exercise 1.1.16

\(x=\sqrt{1−y^2}\)

Exercise 1.1.17

\(y=\dfrac{3x+5}{7x−1}\)

Solution

function

Exercise 1.1.18

\(x^2+y^2=9\)

Exercise 1.1.19

\(2xy=1\)

Solution

function

Exercise 1.1.20

\(x=y^3\)

Exercise 1.1.21

\(y=x^3\)

Solution

function

Exercise 1.1.22

\(y=\sqrt{1−x^2}\)

Exercise 1.1.23

\(x=\pm\sqrt{1-y}\)

Solution

function

Exercise 1.1.24

\(y=\pm\sqrt{1−x}\)

Exercise 1.1.25

\(y^2=x^2\)

Solution

not a function

Exercise 1.1.26

\(y^3=x^2\)

For the following exercises, evaluate the function f at the indicated values \(f(−3),f(2),f(−a),−f(a),f(a+h)\).

Exercise 1.1.27

\(f(x)=2x−5\)

Solution

\(f(−3)=−11\);    \(f(2)=−1\);    \(f(−a)=−2a−5\);    \(−f(a)=−2a+5\);    \(f(a+h)=2a+2h−5\)

Exercise 1.1.28

\(f(x)=−5x^2+2x−1\)

Exercise 1.1.29

\(f(x)=\sqrt{2−x}+5\)

Solution

\(f(−3)=\sqrt{5+5}\);    \(f(2)=5\);    \(f(−a)=\sqrt{2+a+5}\);    \(−f(a)=−\sqrt{2−a}−5\); \(f(a+h)=\sqrt{2−a−h}+5\)

Exercise 1.1.30

\(f(x)=\dfrac{6x−1}{5x+2}\)

Exercise 1.1.31

\(f(x)=|x−1|−|x+1|\)

Solution

\(f(−3)=2\);    \(f(2)=1−3=−2\);    \(f(−a)=|−a−1|−|−a+1|\);    \(−f(a)=−|a−1| +|a+1|\);    \(f(a+h)= |a+h−1|−|a+h+1|\)

Exercise 1.1.32

Given the function \(g(x)=5−x^2\), evaluate \(g(x+h)−g(x)h\), \(h{\neq}0\).

Exercise 1.1.33

Given the function \(g(x)=x^2+2x\), evaluate \(\dfrac{g(x)−g(a)}{x−a}\), \(x{\neq}a\).

Solution

\(\dfrac{g(x)−g(a)}{x−a}=x+a+2\), \(x{\neq}a\)

Exercise 1.1.34

Given the function \(k(t)=2t−1\):

a. Evaluate \(k(2)\).
b. Solve \(k(t)=7\).

Exercise 1.1.35

Given the function \(f(x)=8−3x\):

a. Evaluate \(f(−2)\).
b. Solve \(f(x)=−1\).

Solution

a. \(f(−2)=14\); b. \(x=3\)

Exercise 1.1.36

Given the function \(p(c)=c^2+c\):

a. Evaluate \(p(−3)\).
b. Solve \(p(c)=2\).

Exercise 1.1.37

Given the function \(f(x)=x^2−3x\):

a. Evaluate \(f(5)\).
b. Solve \(f(x)=4\).

Solution

a. \(f(5)=10\); b. \(x=−1\) or \(x=4\)

Exercise 1.1.38

Given the function \(f(x)=\sqrt{x+2}\):

a. Evaluate \(f(7)\).
b. Solve \(f(x)=4\).

Exercise 1.1.39

Consider the relationship \(3r+2t=18\).

a. Write the relationship as a function \(r=f(t)\).
b. Evaluate \(f(−3)\).
c. Solve \(f(t)=2\).

Solution

a. \(f(t)=6−\frac{2}{3}t\); b. \(f(−3)=8\); c. \(t=6\)

Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

Exercise 1.1.40

Graph of relation.

Exercise 1.1.41

Graph of relation.

Solution

not a function

Exercise 1.1.42

Graph of relation.

Exercise 1.1.43

Graph of relation.

Solution

function

Exercise 1.1.44

Graph of relation.

Exercise 1.1.45

Graph of relation.

Solution

function

Exercise 1.1.46

Graph of relation.

Exercise 1.1.47

  Graph of relation.

Solution

function

Exercise 1.1.48

Graph of relation.

Exercise 1.1.49

Graph of relation.

Solution

function

Exercise 1.1.50

Graph of relation.

Exercise 1.1.51

Graph of relation.

Solution

function

Exercise 1.1.52

Given the following graph,

  • Evaluate \(f(−1)\).
  • Solve for \(f(x)=3\).

Graph of relation.

Exercise 1.1.53

Given the following graph,

  • Evaluate \(f(0)\).
  • Solve for \(f(x)=−3\).

Graph of relation.
 

Solution

a.\(f(0)=1\); b. \(f(x)=−3\), \(x=−2\) or \(x=2\)

Exercise 1.1.54

Given the following graph,

  • Evaluate \(f(4)\).
  • Solve for \(f(x)=1\).

Graph of relation.
 

For the following exercises, determine if the given graph is a one-to-one function.

Exercise 1.1.55

Graph of a circle.

Solution

not a function so it is also not a one-to-one function

Exercise 1.1.56

Graph of a parabola.

Exercise 1.1.57

Graph of a rotated cubic function.

Solution

one-to- one function

Exercise 1.1.58

Graph of half of 1/x.

Exercise 1.1.59

Graph of a function

Solution

function, but not one-to-one

Numeric

For the following exercises, determine whether the relation represents a function.

Exercise 1.1.60

\(\{(−1,−1),(−2,−2),(−3,−3)\}\)

Exercise 1.1.61

\(\{(3,4),(4,5),(5,6)\}\)

Solution

function

Exercise 1.1.62

\((2,5),(7,11),(15,8),(7,9)\}\)

For the following exercises, determine if the relation represented in table form represents \(y\) as a function of \(x\).

Exercise 1.1.63

\(x\) 5 10 15
\(y\) 3 8 14
Solution

function

Exercise 1.1.64

\(x\) 5 10 15
\(y\) 3 8 8

Exercise 1.1.65

\(x\) 5 10 10
\(y\) 3 8 14

Solution

not a function

For the following exercises, use the function f represented in Table 1.1.14.

\(x\) \(f(x)\)
0 74
1 28
2 1
3 53
4 56
5 3
6 36
7 45
8 14
9 47

Table 1.1.14

Exercise 1.1.66

Evaluate \(f(3)\).

Exercise 1.1.67

Solve \(f(x)=1\).

Solution

\(f(x)=1\), \(x=2\)

For the following exercises, evaluate the function \(f\) at the values \(f(−2)\), \(f(−1)\), \(f(0)\), \(f(1)\),and \(f(2)\).

Exercise 1.1.68

\(f(x)=4−2x\)

Exercise 1.1.69

\(f(x)=8−3x\)

Solution

\(f(−2)=14\);\(f(−1)=11\);\(f(0)=8\);\(f(1)=5\);\(f(2)=2\)

Exercise 1.1.70

\(f(x)=8x^2−7x+3\)

Exercise 1.1.71

\(f(x)=3+\sqrt{x+3}\)

Solution

\(f(−2)=4\); \(f(−1)=4.414\); \(f(0)=4.732\); \(f(1)=4.5\); \(f(2)=5.236\)

Exercise 1.1.72

\(f(x)=\dfrac{x-2}{x+3}\)

Exercise 1.1.73

\(f(x)=3x\)

Solution

\(f(−2)=19\); \(f(−1)=13\); \(f(0)=1\); \(f(1)=3\); \(f(2)=9\)

For the following exercises, evaluate the expressions, given functions \(f\), \(g\), and \(h\):

  • \(f(x)=3x−2\)
  • \(g(x)=5−x^2\)
  • \(h(x)=−2x^2+3x−1\)

Exercise 1.1.74

\(3f(1)−4g(−2)\)

Exercise 1.1.75

\(f\Big(\dfrac{7}{3}\Big)−h(−2)\)

Solution

20

Technology

For the following exercises, graph \(y=x^2\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

Exercise 1.1.76

\([−0.1, 0.1]\)

Exercise 1.1.77

\([−10, 10]\)

Solution

\([0, 100]\)

Graph of a parabola.

Exercise 1.1.78

\([−100,100]\)

For the following exercises, graph \(y=x^3\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

Exercise 1.1.79

\([−0.1, 0.1]\)

Solution

\([−0.001, 0.001]\)

Graph of a cubic function.

Exercise 1.1.80

\([−10, 10]\)

Exercise 1.1.81

\([−100, 100]\)

Solution

\([−1,000,000, 1,000,000]\)

Graph of a cubic function.

For the following exercises, graph \(y=\sqrt{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

Exercise 1.1.82

\([0, 0.01]\)

Exercise 1.1.83

\([0, 100]\)

Solution

\([0, 10]\)

Graph of a square root function.

Exercise 1.1.84

\([0, 10,000]\)

For the following exercises, graph \(y=\sqrt[3]{x}\) on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

Exercise 1.1.85

\([−0.001,0.001]\)

Solution

\([−0.1,0.1]\)

Graph of a square root function.

Exercise 1.1.86

\([−1000,1000]\)

Exercise 1.1.87

\([−1,000,000,1,000,000]\)

Solution

\([−100, 100]\)

Graph of a cubic root function.

Real-World Applications

Exercise 1.1.88

The amount of garbage, \(G\), produced by a city with population \(p\) is given by \(G=f(p)\). \(G\) is measured in tons per week, and p is measured in thousands of people.

The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\).Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\).
Explain the meaning of the statement \(f(5)=2\).

Exercise 1.1.89

The number of cubic yards of dirt, \(D\), needed to cover a garden with area a square feet is given by \(D=g(a)\).

a. A garden with area 5000 \(ft^2\) requires 50 yd3 of dirt. Express this information in terms of the function \(g\).
b. Explain the meaning of the statement \(g(100)=1\).

Solution

a. \(g(5000)=50\); b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

Exercise 1.1.90

Let \(f(t)\) be the number of ducks in a lake \(t\) years after 1990. Explain the meaning of each statement:

a. f(5)=30
b. f(10)=40

Exercise 1.1.91

Let \(h(t)\) be the height above ground, in feet, of a rocket \(t\) seconds after launching. Explain the meaning of each statement:

a. \(h(1)=200\)
b. \(h(2)=350\)

Solution

a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.

Exercise 1.1.92

Show that the function \(f(x)=3(x−5)2+7\) is not one-to-one.

1.2: Domain and Range

In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions.

Section Exercises

Verbal

Exercise 1.2.1

Why does the domain differ for different functions?

Solution

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

Exercise 1.2.2

How do we determine the domain of a function defined by an equation?

Exercise 1.2.3

Explain why the domain of \(f(x)=\sqrt[3]{x}\) is different from the domain of \(f(x)=\sqrt{x}\).

Solution

There is no restriction on x for \(f(x)=\sqrt[3]{x}\) because you can take the cube root of any real number. So the domain is all real numbers, \((−∞,∞)\). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x-values are restricted for \(f(x)=\sqrt{x}\) to nonnegative numbers and the domain is \([0,∞)\).

Exercise 1.2.4

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

Exercise 1.2.5

How do you graph a piecewise function?

Solution: Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate \(−∞\) or \(∞\). Combine the graphs to find the graph of the piecewise function.

Algebraic

For the following exercises, find the domain of each function using interval notation.

Exercise 1.2.6

\(f(x)=−2x(x−1)(x−2)\)

Exercise 1.2.7

\(f(x)=5−2x^2\)

Solution

\((-\infty,\infty)\)

Exercise 1.2.8

\(f(x)=3\sqrt{x-2}\)

Exercise 1.2.9

 

\(f(x)=3−\sqrt{6−2x}\)

Solution

\(\left(-\infty,3\right]\)

Exercise 1.2.10

\(f(x)=\sqrt{4-3x}\)

Exercise 1.2.11

\(f(x)=\sqrt{x^2+4}\)

Solution

\((-\infty,\infty)\)

Exercise 1.2.12

\(f(x)=\sqrt[3]{1-2x}\)

Exercise 1.2.13

\(f(x)=\sqrt[3]{x-1}\)

Solution

\((-\infty,\infty)\)

Exercise 1.2.14

\(f(x)=\dfrac{9}{x-6}\)

Exercise 1.2.15

\(f(x)=\dfrac{3x+1}{4x+2}\)

Solution

\((-\infty,-\dfrac{1}{2})\cup(-\dfrac{1}{2},\infty)\)

Exercise 1.2.16

\(f(x)=\dfrac{\sqrt{x+4}}{x-4}\)

Exercise 1.2.17

\(f(x)=\dfrac{x-3}{x^2+9x-22}\)

Solution

\((-\infty,-11)\cup(-11,2)\cup(2,\infty)\)

Exercise 1.2.18

\(f(x)=\dfrac{1}{x^2-x-6}\)

Exercise 1.2.19

\(f(x)=\dfrac{2x^3−250}{x^2−2x−15}\)

Solution

\((-\infty,-3)\cup(-3,5)\cup(5,\infty)\)

Exercise 1.2.20

\(\dfrac{5}{\sqrt{x-3}}\)

Exercise 1.2.21

\(\dfrac{2x+1}{\sqrt{5-x}}\)

Solution

\((-\infty,5)\)

Exercise 1.2.22

\(\dfrac{\sqrt{x-4}}{\sqrt{x-6}}\)

Exercise 1.2.23

\(\dfrac{\sqrt{x-6}}{\sqrt{x-4}}\)

Solution

\(\left[6,\infty\right)\)

Exercise 1.2.24

\(f(x)=\dfrac{x}{x}\)

Exercise 1.2.25

\(f(x)=\dfrac{x^2-9x}{x^2-81}\)

Solution

\((-\infty,-9)\cup(-9,9)\cup(9,\infty)\)

Exercise 1.2.26

Find the domain of the function \(f(x)=\sqrt{2x^3-50x}\) by:

a. using algebra

b. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is nonnegative..

Graphical

For the following exercises, write the domain and range of each function using interval notation.

Exercise 1.2.27

Graph of a function from \(\left(2, 8\right]\).

Solution

domain: \(\left(2,8\right]\), range \(\left[6,8\right)\)

Exercise 1.2.28

Graph of a function from \(\left[4, 8\right)\).

Exercise 1.2.29

Graph of a function [-4,4]

Solution

domain: \([−4, 4]\), range: \([0, 2]\)

Exercise 1.2.30

Graph of a function [2,6]

Exercise 1.2.31

Graph of a function [-5,3)

Solution

domain: \(\left[−5, 3\right)\), range: \([0,2]\)

Exercise 1.2.32

Graph of a function from [-3, 2).

Exercise 1.2.33

Graph of a function from (-infinity, 2].

Solution

domain: \(\left(−\infty,1\right]\), range: \(\left[0,\infty\right)\)

Exercise 1.2.34

Graph of a function from [-4, infinity).

Exercise 1.2.35

Graph of a function from [-6, -1/6]U[1/6, 6]/.

Solution

domain: \([−6,−16]\cup[16,6]\); range: \([−6,−16]\cup[16,6]\)

Exercise 1.2.36

Graph of a function from (-2.5, infinity).

Exercise 1.2.37

Graph of a function from [-3, infinity).

Solution

domain: \(\left[−3, \infty\right)\); range: \(\left[0,\infty\right)\)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

Exercise 1.2.38

\(f(x)= \begin{cases}  x+1 & \text{if $x < -2$}  \\ -2x-3 & \text{if $x {\geq} -2$} \end{cases}\)

Exercise 1.2.39

\(f(x)= \begin{cases}  2x-1 & \text{if $x < 1$}  \\ 1+x & \text{if $x {\geq} 1$} \end{cases}\)

Solution

domain: \((−\infty,\infty)\)

Graph of f(x).

Exercise 1.2.40

\(f(x)= \begin{cases}  x+1 & \text{if $x < 0$}  \\ x-1 & \text{if $x > 0$} \end{cases}\)

Exercise 1.2.41

\(f(x)= \begin{cases}  3 & \text{if $x < 0$}  \\ \sqrt{x} & \text{if $x {\geq} 0$} \end{cases}\)

Solution

domain: \((−\infty,\infty)\)

Graph of f(x).

 

Exercise 1.2.42

\(f(x)= \begin{cases}  x^2 & \text{if $x < 0$}  \\ 1-x & \text{if $x > 0$} \end{cases}\)

Exercise 1.2.43

\(f(x)= \begin{cases}  x^2 & \text{if $x < 0$}  \\ x+2 & \text{if $x {\geq} 0$} \end{cases}\)

Solution

domain: \((−\infty,\infty)\)

Graph of f(x).

Exercise 1.2.44

\(f(x)= \begin{cases}  x+1 & \text{if $x < 1$}  \\ x^3 & \text{if $x {\geq} 1$} \end{cases}\)

Exercise 1.2.45

\(f(x)= \begin{cases}  |x| & \text{if $x < 2$}  \\ 1 & \text{if $x {\geq} 2$} \end{cases}\)

Solution

domain: \((−\infty,\infty)\)

Graph of f(x).

Numeric

For the following exercises, given each function \(f\),evaluate \(f(−3)\), \(f(−2)\), \(f(−1)\), and \(f(0)\).

Exercise 1.2.46

\(f(x)= \begin{cases}  x+1 & \text{if $x < -2$}  \\ -2x-3 & \text{if $x {\geq} -2$} \end{cases}\)

Exercise 1.2.47

\(f(x)= \begin{cases}  1 & \text{if $x \leq -3$}  \\ 0 & \text{if $x > -3$} \end{cases}\)

Solution

\(f(−3)=1\); \(f(−2)=0\); \(f(−1)=0\); \(f(0)=0\)

Exercise 1.2.48

\(f(x)= \begin{cases}  -2x^2+3 & \text{if $x \leq -1$}  \\ 5x-7 & \text{if $x > -1$} \end{cases}\)

For the following exercises, given each function \(f\), evaluate \(f(−1)\), \(f(0)\), \(f(2)\), and \(f(4)\).

Exercise 1.2.49

\(f(x)= \begin{cases}  7x+3 & \text{if $x < 0$}  \\ 7x+6 & \text{if $x {\geq} 0$} \end{cases}\)

Solution

\(f(−1)=−4\); \(f(0)=6\); \(f(2)=20\); \(f(4)=34\)

Exercise 1.2.50

\(f(x)= \begin{cases} x^2-2  & \text{if $x < 2$}  \\ 4+|x-5| & \text{if $x {\geq} 2$} \end{cases}\)

Exercise 1.2.51

\(f(x)= \begin{cases} 5x  & \text{if $x < 0$}  \\ 3 & \text{if $0 {\geq} x {\leq} 2$} \\  x^2  & \text{if $x > 3$} \end{cases}\)

Solution

\(f(−1)=−5\); \(f(0)=3\); \(f(2)=3\); \(f(4)=16\)

For the following exercises, write the domain for the piecewise function in interval notation.

Exercise 1.2.52

\(f(x)= \begin{cases} x+1  & \text{if $x < -2$}  \\ -2x-3 & \text{if $x {\geq} -2$} \end{cases}\)

Exercise 1.2.53

\(f(x)= \begin{cases} x^2-2  & \text{if $x < 1$}  \\ -x^2+2 & \text{if $x > 1$} \end{cases}\)

Solution

domain: \((−\infty,1)\cup(1,\infty)\)

Exercise 1.2.54

\(f(x)= \begin{cases} x^2-3  & \text{if $x < 0$}  \\ -3x^2 & \text{if $x {\geq} 2$} \end{cases}\)

Technology

Exercise 1.2.55

Graph \(y=\dfrac{1}{x^2}\) on the viewing window \([−0.5,−0.1]\) and \([0.1,0.5]\). Determine the corresponding range for the viewing window. Show the graphs.

Solution

Graph of the equation from [-0.5, -0.1].

window: \([−0.5,−0.1]\); range: \([4, 100]\)

Graph of the equation from [0.1, 0.5].

window: \([0.1, 0.5]\); range: \([4, 100]\)

Exercise 1.2.56

Graph \(y=\dfrac{1}{x}\) on the viewing window \([−0.5,−0.1]\) and \([0.1, 0.5]\). Determine the corresponding range for the viewing window. Show the graphs.

Extension

Exercise 1.2.57

Suppose the range of a function \(f\) is \([−5, 8]\). What is the range of \(|f(x)|\)?

Solution

\([0, 8]\)

Exercise 1.2.58

Create a function in which the range is all nonnegative real numbers.

Exercise 1.2.59

Create a function in which the domain is \(x>2\).

Solution

Many answers. One function is \(f(x)=\dfrac{1}{\sqrt{x-2}}\).

Real-World Applications

Exercise 1.2.60

The height \(h\) of a projectile is a function of the time \(t\) it is in the air. The height in feet for \(t\) seconds is given by the function \(h(t)=−16t^2+96t\). What is the domain of the function? What does the domain mean in the context of the problem?

Solution

The domain is \([0, 6]\); it takes 6 seconds for the projectile to leave the ground and return to the ground

Exercise 1.2.61

The cost in dollars of making \(x\) items is given by the function \(C(x)=10x+500\).

a. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.

b. What is the cost of making 25 items?

c. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, \(C(x)\)?

1.3: Rates of Change and Behavior of Graphs

In this section, we will investigate changes in functions. For example, a rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. Identifying points that mark the interval on a graph can be used to find the average rate of change. Comparing pairs of input and output values in a table can also be used to find the average rate of change.

Section Exercises

Verbal

Exercise 1.3.1

Can the average rate of change of a function be constant?

Solution

Yes, the average rate of change of all linear functions is constant.

Exercise 1.3.2

If a function \(f\) is increasing on \((a,b)\) and decreasing on \((b,c)\), then what can be said about the local extremum of \(f\) on \((a,c)\)?

Exercise 1.3.3

How are the absolute maximum and minimum similar to and different from the local extrema?

Solution

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

Exercise \(\PageIndex{4}\)

How does the graph of the absolute value function compare to the graph of the quadratic function, \(y=x^2\), in terms of increasing and decreasing intervals?

 

Algebraic

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h.

Exercise \(\PageIndex{5}\)

\(f(x)=4x^2−7\) on \([1, b]\)

Solution

\(4(b+1)\)

Exercise \(\PageIndex{6}\)

\(g(x)=2x^2−9\) on \([4, b]\)

Exercise \(\PageIndex{7}\)

\(p(x)=3x+4\) on \([2, 2+h]\)

Solution

3

Exercise \(\PageIndex{8}\)

\(k(x)=4x−2\) on \([3, 3+h]\)

Exercise \(\PageIndex{9}\)

\(f(x)=2x^2+1\) on \([x,x+h]\)

Solution

\(4x+2h\)

Exercise \(\PageIndex{10}\)

\(g(x)=3x^2−2\) on \([x,x+h]\)

Exercise \(\PageIndex{11}\)

\(a(t)=\dfrac{1}{t+4}\) on \([9,9+h]\)

Solution

\(\dfrac{−1}{13(13+h)}\)

Exercise \(\PageIndex{12}\)

\(b(x)=1x+3\) on \([1,1+h]\)

Exercise \(\PageIndex{13}\)

\(j(x)=3x^3\) on \([1,1+h]\)

Solution

\(3h^2+9h+9\)

Exercise \(\PageIndex{14}\)

\(r(t)=4t^3\) on \([2,2+h]\)

Exercise \(\PageIndex{15}\)

\(\frac{f(x+h)−f(x)}{h}\) given \(f(x)=2x^2−3x\) on \([x,x+h]\)

Solution

\(4x+2h−3\)

Graphical

For the following exercises, consider the graph of f shown in Figure \(\PageIndex{15}\).

Graph of a polynomial.

Figure \(\PageIndex{15}\): Graph of a polynomial.

Exercise \(\PageIndex{16}\)

Estimate the average rate of change from \(x=1\) to \(x=4\).

Exercise \(\PageIndex{17}\)

Estimate the average rate of change from \(x=2\) to \(x=5\).

Solution

\(\dfrac{4}{3}\)

Exercise \(\PageIndex{18}\)

Graph of an absolute function.

Exercise \(\PageIndex{19}\)

Graph of a cubic function.

Solution

increasing on \((−\infty,−2.5)\cup(1,\infty)\), decreasing on \((−2.5, 1)\)

Exercise \(\PageIndex{20}\)

Graph of a cubic function

Exercise \(\PageIndex{21}\)

Graph of a reciprocal function.

Solution

increasing on \((−\infty,1)\cup(3,4)\), decreasing on \((1,3)\cup(4,\infty)\)

 

For the following exercises, consider the graph shown in Figure \(\PageIndex{15}\).

Figure \(\PageIndex{15}\): Graph of a cubic function.

Exercise \(\PageIndex{22}\)

Estimate the intervals where the function is increasing or decreasing.

Exercise \(\PageIndex{23}\)

Estimate the point(s) at which the graph of f has a local maximum or a local minimum.

Solution

local maximum: \((−3, 60)\), local minimum: \((3, −60)\)

For the following exercises, consider the graph in Figure \(\PageIndex{16}\).

Graph of a cubic function.

Figure \(\PageIndex{16}\): Graph of a cubic function.

Exercise 1.3.24

If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.

Exercise 1.3.25

If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.

Solution

absolute maximum at approximately \((7, 150)\), absolute minimum at approximately \((−7.5, −220)\)

Numberic

Exercise 1.3.26

Table 1.3.3 gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

Year Sales (millions of dollars)
1998 201
1999 219
2000 233
2001 243
2002 249
2003 251
2004 249
2005 243
2006 233

Exercise 1.3.27

Table \(\PageIndex{4}\) gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year Population (thousands)
2000 87
2001 84
2002 83
2003 80
2004 77
2005 76
2006 78
2007 81
2008 85

Solution

a. –3000; b. –1250

For the following exercises, find the average rate of change of each function on the interval specified.

Exercise 1.3.28

\(f(x)=x^2\) on \([1, 5]\)

Exercise 1.3.29

\(h(x)=5−2x^2\) on \([−2,4]\)

Solution

-4

Exercise 1.3.30

\(q(x)=x^3\) on \([−4,2]\)

Exercise 1.3.31

\(g(x)=3x^3−1\) on \([−3,3]\)

Solution

27

Exercise 1.3.32

\(y=\dfrac{1}{x}\) on \([1, 3]\)

Exercise 1.3.33

\(p(t)=\dfrac{(t^2-4)(t+1)}{t^2+3}\) on \([-3,1]\)

Solution

–0.167

Exercise 1.3.34

\(k(t)=6t^2+\dfrac{4}{t^3}\) on \([−1,3]\)

Technology

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

Exercise 1.3.35

\(f(x)=x^4−4x^3+5\)

Solution

Local minimum at \((3,−22)\), decreasing on \((−\infty, 3)\), increasing on \((3, \infty)\)

Exercise 1.3.36

\(h(x)=x^5+5x^4+10x^3+10x^2−1\)

Exercise 1.3.37

\(g(t)=t\sqrt{t+3}\)

Solution

Local minimum at \((−2,−2)\), decreasing on \((−3,−2)\), increasing on \((−2, \infty)\)

Exercise 1.3.38

A common method for preparing oxygen is the decompositio

Solution

\(k(t)=3t^{\frac{2}{3}}−t\)

Exercise 1.3.39

\(m(x)=x^4+2x^3−12x^2−10x+4\)

Solution

Local maximum at \((−0.5, 6)\), local minima at \((−3.25,−47)\) and \((2.1,−32)\), decreasing on \((−\infty,−3.25)\) and \((−0.5, 2.1)\), increasing on \((−3.25, −0.5)\) and \((2.1,\infty)\)

Exercise \(\PageIndex{4}\)0

\(n(x)=x^4−8x^3+18x^2−6x+2\)

Extension

Exercise \(\PageIndex{4}\)1

The graph of the function \(f\) is shown in Figure \(\PageIndex{17}\)

Graph of f(x) on a graphing calculator.

Figure \(\PageIndex{17}\): Graph of f(x) on a graphing calculator.

Based on the calculator screen shot, the point \((1.333, 5.185)\) is which of the following?

A.  a relative (local) maximum of the function
B.  the vertex of the function
C.  the absolute maximum of the function
D.  a zero of the function

Solution

A

Exercise \(\PageIndex{4}\)2

Let \(f(x)=\dfrac{1}{x}\). Find a number \(c\) such that the average rate of change of the function f on the interval \((1,c)\) is \(-\dfrac{1}{4}\).

Exercise \(\PageIndex{4}\)3

Let \(f(x)=\dfrac{1}{x}\). Find the number \(b\) such that the average rate of change of f on the interval \((2,b)\) is \(−\dfrac{1}{10}\).

Solution

\(b=5\)

Real-World Applications

Exercise \(\PageIndex{4}\)4

At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?

Exercise \(\PageIndex{4}\)5

A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

Solution

2.7 gallons per minute

Exercise \(\PageIndex{4}\)6

Near the surface of the moon, the distance that an object falls is a function of time. It is given by \(d(t)=2.6667t^2\), where \(t\) is in seconds and \(d(t)\) is in feet. If an object is dropped from a certain height, find the average velocity of the object from \(t=1\) to \(t=2\).

Exercise \(\PageIndex{4}\)7

The graph in Figure \(\PageIndex{18}\) illustrates the decay of a radioactive substance over \(t\) days.

Graph of an exponential function.

Figure \(\PageIndex{18}\): Graph of an exponential function.

Use the graph to estimate the average decay rate from \(t=5\) to \(t=15\).

Solution

approximately –0.6 milligrams per day

1.4: Composition of Functions

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. The cost depends on the temperature, and the temperature depends on the day.  By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

Section Exercises

Verbal

Exercise 1.4.1

How does one find the domain of the quotient of two functions, \(\frac{f}{g}\)?

Solution

Find the numbers that make the function in the denominator \(g\) equal to zero, and check for any other domain restrictions on \(f\) and \(g\), such as an even-indexed root or zeros in the denominator.

Exercise 1.4.2

What is the composition of two functions, \(f{\circ}g\)?

Exercise 1.4.3

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

Solution

Yes. Sample answer: Let \(f(x)=x+1\) and \(g(x)=x−1\). Then \(f(g(x))=f(x−1)=(x−1)+1=x\) and \(g(f(x))=g(x+1)=(x+1)−1=x\). So \(f{\circ}g=g{\circ}f\).

Exercise 1.4.4

How do you find the domain for the composition of two functions, \(f{\circ}g\)?

Algebraic

Exercise 1.4.5

Given \(f(x)=x^2+2x\) and \(g(x)=6−x^2\),find \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\).Determine the domain for each function in interval notation.

Solution

\((f+g)(x)=2x+6\), domain: \((−\infty,\infty)\)

\((f−g)(x)=2x^2+2x−6\), domain: \((−\infty,\infty)\)

\((fg)(x)=−x^4−2x^3+6x^2+12x\), domain: \((−\infty,\infty)\)

\(\Big(\dfrac{f}{g}\Big)(x)=\dfrac{x^2+2x}{6−x^2},\) domain: \( (−\infty,−\sqrt{6})\cup(\sqrt{6},\sqrt{6})\cup(\sqrt{6},\infty)\)

 

Exercise 1.4.6

Given \(f(x)=−3x^2+x\) and \(g(x)=5\), find \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\). Determine the domain for each function in interval notation.

Exercise 1.4.7

Given \(f(x)=2x^2+4x\) and \(g(x)=\frac{1}{2x}\),find \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\). Determine the domain for each function in interval notation.

Solution

\((f+g)(x)=\frac{4x^3+8x^2+1}{2x}\), domain: \((−\infty,0)\cup(0,\infty)\)

\((f−g)(x)=\frac{4x3+8x2−1}{2x}\), domain: \((−\infty,0)\cup(0,\infty)\)

\((fg)(x)=x+2\), domain: \((−\infty,0)\cup(0,\infty)\)

\((fg)(x)=4x^3+8x^2\), domain: \((−\infty,0)\cup(0,\infty)\)

Exercise 1.4.8

Given \(f(x)=\dfrac{1}{x−4}\) and \(g(x)=\frac{1}{6−x}\), find\(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\). Determine the domain for each function in interval notation.

Exercise 1.4.9

Given \(f(x)=3x^2\) and \(g(x)=\sqrt{x−5}\), find \(f+g\), \(f−g\), \(fg\), and \(\frac{f}{g}\). Determine the domain for each function in interval notation.

Solution

\((f+g)(x)=3x^2+\sqrt{x−5}\), domain: \(\left[5,\infty\right)\)

\((f−g)(x)=3x^2−\sqrt{x−5}\), domain: \(\left[5,\infty\right)\)

\((fg)(x)=3x^2\sqrt{x−5}\), domain: \(\left[5,\infty\right)\)

\(\Big(\dfrac{f}{g}\Big)(x)=\frac{3x^2}{\sqrt{x−5}}\), domain: \((5,\infty)\)

Exercise 1.4.10

Given \(f(x)=\sqrt{x}\) and \(g(x)=|x−3|\), find \(\frac{g}{f}\). Determine the domain of the function in interval notation.

Exercise 1.4.11

Given \(f(x)=2x^2+1\) and \(g(x)=3x−5\), find the following:

a. \(f(g(2))\)
b. \(f(g(x))\)
c. \(g(f(x))\)
d. \((g{\circ}g)(x)\)
e. \((f{\circ}f)(−2)\)

Solution

a. 3; b. \(f(g(x))=2(3x−5)^2+1\); c. \(f(g(x))=6x^2−2\); d. \((f{\circ}g)(x)=3(3x−5)−5=9x−20\); e. \((f{\circ}f)(−2)=163\)

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers.

Exercise 1.4.12

\(f(x)=x^2+1\), \(g(x)=\sqrt{x+2}\)

Exercise 1.4.13

\(f(x)=\sqrt{x}+2\), \(g(x)=x^2+3\)

Solution

\(f(g(x))=\sqrt{x+3}\), \(g(f(x))=x+4\sqrt{x}+7\)

Exercise 1.4.14

\(f(x)=|x|\), \(g(x)=5x+1\)

Exercise 1.4.15

\(f(x)=\sqrt[3]{x}\), \(g(x)=\frac{x+1}{x^3}\)

Solution

\(f(g(x))=\sqrt[3]{\dfrac{x+1}{x^3}}=\dfrac{\sqrt[3]{x+1}}{x}\), \(g(f(x))=\dfrac{\sqrt[3]{x}+1}{x}\)

Exercise 1.4.16

\(f(x)=\dfrac{1}{x−6}\), \(g(x)=\dfrac{7}{x}+6\)

Exercise 1.4.17

\(f(x)=\dfrac{1}{x−4}\), \(g(x)=\dfrac{2}{x}+4\)

Solution

\((f{\circ}g)(x)=\dfrac{1}{\frac{2}{x}+4-4}=\dfrac{x}{2}\), \((g{\circ}f)(x)=2x-4\)

For the following exercises, use each set of functions to find f(g(h(x))). Simplify your answers.

Exercise 1.4.18

\(f(x)=x^4+6\), \(g(x)=x−6\), and \(h(x)=\sqrt{x}\)

Exercise 1.4.19

\(f(x)=x^2+1\), \(g(x)=\frac{1}{x}\), and \(h(x)=x+3\)

Solution

\(f(g(h(x)))=\Big(\dfrac{1}{x+3}\Big)^2+1\)

Exercise 1.4.20

Given \(f(x)=\frac{1}{x}\) and \(g(x)=x−3\), find the following:

a. \((f{\circ}g)(x)\)
b. the domain of \((f{\circ}g)(x)\) in interval notation
c. \((g{\circ}f)(x)\)
d. the domain of \((g{\circ}f)(x)\)
3. \(\Big(\frac{f}{g}\Big)x\)

Exercise 1.4.21

Given \(f(x)=\sqrt{2−4x}\) and \(g(x)=−\frac{3}{x}\), find the following:

a. \((g{\circ}f)(x)\)
b. the domain of \((g{\circ}f)(x)\) in interval notation

Solution

a. \((g{\circ}f)(x)=−\dfrac{3}{\sqrt{2−4x}}\); b. \((−\infty,\frac{1}{2})\)

Exercise 1.4.22

Given the functions \(f(x)=\frac{1−x}{x}\) and \(g(x)=\frac{1}{1+x^2}\), find the following:

a. \((g{\circ}f)(x)\)
b. \((g{\circ}f)(2)\)

Exercise 1.4.23

Given functions \(p(x)=\frac{1}{\sqrt{x}}\) and \(m(x)=x^2−4\), state the domain of each of the following functions using interval notation:

a. \(\frac{p(x)}{m(x)}\)
b. \(p(m(x))\)
c. \(m(p(x))\)

Solution

a. \((0,2)\cup(2,\infty)\); b. \((−\infty,−2)\cup(2,\infty)\); c. \((0,\infty)\)

Exercise 1.4.24

Given functions \(q(x)=\frac{1}{\sqrt{x}}\) and \(h(x)=x^2−9\), state the domain of each of the following functions using interval notation.

a. \(\frac{q(x)}{h(x)}\)
b. \(q(h(x))\)
c. \(h(q(x))\)

Exercise 1.4.25

For \(f(x)=\frac{1}{x}\) and \(g(x)=\sqrt{x−1}\), write the domain of \((f{\circ}g)(x)\) in interval notation.

Solution

\((1,\infty)\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\).

Exercise 1.4.26

\(h(x)=(x+2)^2\)

Exercise 1.4.27

\(h(x)=(x−5)^3\)

Solution

sample: \(f(x)=x^3\)

\(g(x)=x−5\)

Exercise 1.4.28

\(h(x)=\frac{3}{x−5}\)

Exercise 1.4.29

\(h(x)=\frac{4}{(x+2)^2}\)

Solution

sample:\(f(x)=\frac{4}{x}\)

\(g(x)=(x+2)^2\)

Exercise 1.4.30

\(h(x)=4+\sqrt[3]{x}\)

Exercise 1.4.31

\(h(x)=\sqrt[3]{\frac{1}{2x−3}}\)

Solution

sample:\(f(x)=\sqrt[3]{x}\)

\(g(x)=\frac{1}{2x−3}\)

Exercise 1.4.32

\(h(x)=\frac{1}{(3x^2−4)^{−3}}\)

Exercise 1.4.33

\(h(x)=\sqrt[4]{\frac{3x−2}{x+5}}\)

Solution

sample:\(f(x)=\sqrt[4]{x}\)

\(g(x)=\frac{3x−2}{x+5}\)

Exercise 1.4.34

\(h(x)=(\frac{8+x^3}{8−x^3})^4\)

Exercise 1.4.35

\(h(x)=\sqrt{2x+6}\)

Solution

sample: \(f(x)=\sqrt{x}\)

\(g(x)=2x+6\)

Exercise 1.4.36

\(h(x)=(5x−1)^3\)

Exercise 1.4.37

\(h(x)=\sqrt[3]{x−1}\)

Solution

sample: \(f(x)=\sqrt[3]{x}\)
\(g(x)=(x−1)\)

Exercise 1.4.38

\(h(x)=|x^2+7|\)

Exercise 1.4.39

\(h(x)=\frac{1}{(x−2)^3}\)

Solution

sample: \(f(x)=x^3\)
\(g(x)=\frac{1}{x−2}\)

Exercise 1.4.40

\(h(x)=\Big(\frac{1}{2x−3}\Big)^2\)

Exercise 1.4.41

\(h(x)=\sqrt { \frac{2x−1}{3x+4}}\)

Solution

sample: \(f(x)=\sqrt{x}\)
\(g(x)=\frac{2x−1}{3x+4}\)

Graphical

For the following exercises, use the graphs of \(f\),shown in Figure 1.4.6, and \(g\),shown in Figure 1.4.7, to evaluate the expressions.

Graph of a function.

Figure 1.4.6: Graph of a function.

Graph of a function.
Figure 1.4.7: Graph of a function.

Exercise 1.4.42

\(f(g(3))\)

Exercise 1.4.43

\(f(g(1))\)

Solution

2

Exercise 1.4.44

\(g(f(1))\)

Exercise 1.4.45

\(g(f(0))\)

Solution

5

Exercise 1.4.46

\(f(f(5))\)

Exercise 1.4.47

\(f(f(4))\)

Solution

4

Exercise 1.4.48

\(g(g(2))\)

Exercise 1.4.49

\(g(g(0))\)

Solution

0

For the following exercises, use graphs of \(f(x)\),shown in Figure 1.4.8, \(g(x)\),shown in Figure 1.4.9, and \(h(x)\),shown in Figure 1.4.10, to evaluate the expressions.

Graph of a parabola

Figure 1.4.8: Graph of a parabola.

Graph of a square root function.
Figure 1.4.9: Graph of a square root function.

Graph of an absolute value function.

Figure 1.4.10: Graph of an absolute value function.

Exercise 1.4.50

\(g(f(1))\)

Exercise 1.4.51

\(g(f(2))\)

Solution

2

Exercise 1.4.52

\(f(g(4))\)

Exercise 1.4.53

\(f(g(1))\)

Solution

1

Exercise 1.4.54

\(f(h(2))\)

Exercise 1.4.55

\(h(f(2))\)

Solution

4

Exercise 1.4.56

\(f(g(h(4)))\)

Exercise 1.4.57

\(f(g(f(−2)))\)

Solution

4

Numeric

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.4.3 to evaluate each expression.

\(x\) \(f(x)\) \(g(x)\)
0 7 9
1 6 5
2 5 6
3 8 2
4 4 1
5 0 8
6 2 7
7 1 3
8 9 4
9 3 0

Table 1.4.3

Exercise 1.4.58

\(f(g(8))\)

Exercise 1.4.59

\(f(g(5))\)

Solution

9

Exercise 1.4.60

\(g(f(5))\)

Exercise 1.4.61

\(g(f(3))\)

Solution

4

Exercise 1.4.62

\(f(f(4))\)

Exercise 1.4.63

\(f(f(1))\)

Solution

2

Exercise 1.4.64

\(g(g(2))\)

Exercise 1.4.65

\(g(g(6))\)

Solution

3

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.4.4 to evaluate the expressions.

\(x\) \(f(x)\) \(g(x)\)
-3 11 -8
-2 9 -3
-1 7 0
0 5 1
1 3 0
2 1 -3
3 -1 -8

Exercise 1.4.66

\((f{\circ}g)(1)\)

Exercise 1.4.66

\((f{\circ}g)(2)\)

Solution

11

Exercise 1.4.67

\((g{\circ}f)(2)\)

Exercise 1.4.68

\((g{\circ}f)(3)\)

Solution

0

Exercise 1.4.69

\((g{\circ}g)(1)\)

Exercise 1.4.70

\((f{\circ}f)(3)\)

Solution

7

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\).

Exercise 1.4.71

\(f(x)=4x+8\), \(g(x)=7−x^2\)

Exercise 1.4.72

\(f(x)=5x+7\), \(g(x)=4−2x^2\)

Solution

\(f(g(0))=27\), \(g(f(0))=−94\)

Exercise 1.4.73

\(f(x)=\sqrt{x+4}\), \(g(x)=12−x^3\)

Exercise 1.4.74

\(f(x)=\frac{1}{x+2}\), \(g(x)=4x+3\)

Solution

\(f(g(0))=\frac{1}{5}\), \(g(f(0))=5\)

For the following exercises, use the functions \(f(x)=2x^2+1\) and \(g(x)=3x+5\) to evaluate or find the composite function as indicated.

Exercise 1.4.75

\(f(g(2))\)

Exercise 1.4.76

\(f(g(x))\)

Solution

\(18x^2+60x+51\)

Exercise 1.4.77

\(g(f(−3))\)

Exercise 1.4.78

\((g{\circ}g)(x)\)

Solution

\(g{\circ}g(x)=9x+20\)

Extensions

For the following exercises, use \(f(x)=x^3+1\) and \(g(x)=\sqrt[3]{x−1}\).

Exercise 1.4.79

Find \((f{\circ}g)(x)\) and \((g{\circ}f)(x)\). Compare the two answers.

Exercise 1.4.80

Find \((f{\circ}g)(2)\) and \((g{\circ}f)(2)\).

Solution

2

Exercise 1.4.81

What is the domain of \((g{\circ}f)(x)\)?

Exercise 1.4.82

What is the domain of \((f{\circ}g)(x)\)?

Solution

\((−\infty,\infty)\)

Exercise 1.4.83

Let \(f(x)=\frac{1}{x}\).

a. Find \((f{\circ}f)(x)\).

b. Is \((f{\circ}f)(x)\) for any function \(f\) the same result as the answer to part (a) for any function? Explain.

For the following exercises, let \(F(x)=(x+1)^5\), \(f(x)=x^5\), and \(g(x)=x+1\).

Exercise 1.4.84

True or False: \((g{\circ}f)(x)=F(x)\).

Solution

False

Exercise 1.4.85

True or False: \((f{\circ}g)(x)=F(x)\).

For the following exercises, find the composition when \(f(x)=x^2+2\) for all \(x\geq0\) and \(g(x)=\sqrt{x−2}\).

Exercise 1.4.86

\((f{\circ}g)(6)\); \((g{\circ}f)(6)\)

Exercise 1.4.87

\((f{\circ}g)(6)=6\); \((g{\circ}f)(6)=6\)

Exercise 1.4.88

\((g{\circ}f)(a)\); \((f{\circ}g)(a)\)

Exercise 1.4.89

\((f{\circ}g)(11)\); \((g{\circ}f)(11)\)

Solution

\((f{\circ}g)(11)=11\) , \((g{\circ}f)(11)=11\)

Real-World Applications

Exercise 1.4.90

The function \(D(p)\) gives the number of items that will be demanded when the price is \(p\). The production cost \(C(x)\) is the cost of producing \(x\) items. To determine the cost of production when the price is $6, you would do which of the following?

a. Evaluate \(D(C(6))\).
b. Evaluate \(C(D(6))\).
c. Solve \(D(C(x))=6\).
d. Solve \(C(D(p))=6\).

Exercise 1.4.91

The function \(A(d)\) gives the pain level on a scale of 0 to 10 experienced by a patient with \(d\) milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient’s system after \(t\) minutes is modeled by \(m(t)\). Which of the following would you do in order to determine when the patient will be at a pain level of 4?

a. Evaluate \(A(m(4))\).
b. Evaluate \(m(A(4))\).
c. Solve \(A(m(t))=4\).
d. Solve \(m(A(d))=4\).

Solution

c

Exercise 1.4.92

A store offers customers a 30% discount on the price x of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function \(P(x)\) that computes the final price of the item in terms of the original price \(x\). (Hint: Use function composition to find your answer.)

Exercise 1.4.93

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to \(r(t)=25\sqrt{t+2}\), find the area of the ripple as a function of time. Find the area of the ripple at \(t=2\).

Solution

\(A(t)=\pi(25\sqrt{t+2})^2\) and \(A(2)=\pi(25\sqrt{4})^2=2500\pi\) square inches

Exercise 1.4.94

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula \(r(t)=2t+1\), express the area burned as a function of time, \(t\) (minutes).

Exercise 1.4.95

Use the function you found in the previous exercise to find the total area burned after 5 minutes.

Solution

\(A(5)=\pi(2(5)+1)2=121\pi\) square units

Exercise 1.4.96

The radius \(r\), in inches, of a spherical balloon is related to the volume, \(V\), by \(r(V)=\sqrt[3]{\frac{3V}{4\pi}}\). Air is pumped into the balloon, so the volume after \(t\) seconds is given by \(V(t)=10+20t\).

a. Find the composite function \(r(V(t))\).

b. Find the exact time when the radius reaches 10 inches.

Exercise 1.4.97

The number of bacteria in a refrigerated food product is given by \(N(T)=23T^2−56T+1\), \(3<T<33\), where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by \(T(t)=5t+1.5\), where \(t\) is the time in hours.

Find the composite function \(N(T(t))\).
Find the time (round to two decimal places) when the bacteria count reaches 6752.

Solution

a. \(N(T(t))=23(5t+1.5)^2−56(5t+1.5)+1\); b. 3.38 hours

1.5: Transformation of Functions

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

Section Exercises

Verbal

Exercise 1.5.1

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

Solution

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

Exercise 1.5.2

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

Exercise 1.5.3

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

Solution

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

Exercise 1.5.4

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

Exercise 1.5.5

How can you determine whether a function is odd or even from the formula of the function?

Solution

For a function \(f\), substitute (−x) for (x) in \(f(x)\). Simplify. If the resulting function is the same as the original function, \(f(−x)=f(x)\), then the function is even. If the resulting function is the opposite of the original function, \(f(−x)=−f(x)\), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

Algebraic

Exercise 1.5.6

Write a formula for the function obtained when the graph of \(f(x)=\sqrt{x}\) is shifted up 1 unit and to the left 2 units.

Exercise 1.5.7

Write a formula for the function obtained when the graph of \(f(x)=|x|\) is shifted down 3 units and to the right 1 unit.

Solution

\(g(x)=|x-1|−3\)

Exercise 1.5.8

Write a formula for the function obtained when the graph of \(f(x)=\dfrac{1}{x}\) is shifted down 4 units and to the right 3 units.

Exercise 1.5.9

Write a formula for the function obtained when the graph of \(f(x)=\dfrac{1}{x^2}\) is shifted up 2 units and to the left 4 units.

Solution

\(g(x)=\dfrac{1}{(x+4)^2}+2\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\).

Exercise 1.5.10

\(y=f(x−49)\)

Exercise 1.5.11

\(y=f(x+43)\)

Solution

The graph of \(f(x+43)\) is a horizontal shift to the left 43 units of the graph of \(f\).

Exercise 1.5.12

\(y=f(x+3)\)

Exercise 1.5.13

\(y=f(x−4)\)

Solution

The graph of \(f(x-4)\) is a horizontal shift to the right 4 units of the graph of f.

Exercise 1.5.14

\(y=f(x)+5\)

Exercise 1.5.15

\(y=f(x)+8\)

Solution

The graph of \(f(x)+8\) is a vertical shift up 8 units of the graph of \(f\).

Exercise 1.5.16

\(y=f(x)−2\)

Exercise 1.5.17

\(y=f(x)−7\)

Solution

The graph of \(f(x)−7\) is a vertical shift down 7 units of the graph of \(f\).

Exercise 1.5.18

\(y=f(x−2)+3\)

Exercise 1.5.19

\(y=f(x+4)−1\)

Solution

The graph of \(f(x+4)−1\) is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of \(f\).

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

Exercise 1.5.20

\(f(x)=4(x+1)^2−5\)

Exercise 1.5.21

\(g(x)=5(x+3)^2−2\)

Solution

decreasing on \((−\infty,−3)\) and increasing on \((−3,\infty)\)

Exercise 1.5.22

\(a(x)=\sqrt{−x+4}\)

Exercise 1.5.23

\(k(x)=−3\sqrt{x}−1\)

Solution

decreasing on \((0, \infty)\)

Graphical

For the following exercises, use the graph of \(f(x)=2^x\) shown in Figure 1.5.30 to sketch a graph of each transformation of \(f(x)\).

 Graph of \(f(x)\).

Figure 1.5.30: Graph of \(f(x)\).

Exercise 1.5.24

\(g(x)=2^x+1\)

Exercise 1.5.25

\(h(x)=2x−3\)

Solution

 

Graph of \(k(x)\)

Graph of \(k(x)\)

Exercise 1.5.26

\(w(x)=2^x−1\)

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

Exercise 1.5.27

\(f(t)=(t+1)^2−3\)

Solution

Graph of \(f(t)\).

Graph of \(f(t)\).

Exercise 1.5.28

\(h(x)=|x−1|+4\)

Exercise 1.5.29

\(k(x)=(x−2)^3−1\)

Solution

Graph of \(k(x)\).

Graph of \(k(x)\).

Exercise 1.5.30

\(m(t)=3+\sqrt{t+2}\)

Numeric

Exercise 1.5.31

Tabular representations for the functions \(f\), \(g\), and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).

\(x\)

-2 -1 0 1

2

\(f(x)\) -2 -1 -3 1 2

 

\(x\)

-1 0 1 2 3
\(g(x)\) -2 -1 -3 1 2

 

\(x\)

-2 -1 0 1

2

\(h(x)\) -1 0 -2 2 3

 

Solution

\(g(x)=f(x-1)\), \(h(x)=f(x)+1\)

Exercise 1.5.32

Tabular representations for the functions \(f\), \(g\), and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).

\(x\)

-2 -1 0 1

2

\(f(x)\) -1 -3 4 2 1

 

\(x\)

-3 -2 -1 0 1
\(g(x)\) -1 -3 4 2 1

 

\(x\)

-2 -1 0 1

2

\(h(x)\) -2 -4 3 1 0

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Exercise 1.5.33

Graph of an absolute function.

Graph of an absolute function.

Solution

\(f(x)=|x-3|−2\)

Exercise 1.5.34

Graph of a parabola.

Graph of a parabola.

Exercise 1.5.35

Graph of a square root function.

Graph of a square root function.

 

Solution

\(f(x)=\sqrt{x+3}−1\)

Exercise 1.5.36

Graph of an absolute function.

Graph of an absolute function.

Exercise 1.5.37

Graph of a parabola

Graph of a parabola

Solution

\(f(x)=(x-2)^2\)

Exercise 1.5.38

Graph of a square root function.

Exercise 1.5.39

Graph of an absolute function.

Graph of an absolute function.

Solution

\(f(x)=|x+3|−2\)

Exercise 1.5.40

Graph of a square root function.

Graph of a square root function.

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

Exercise 1.5.41

 

Graph of a square root function.

Graph of a square root function.

Solution

\(f(x)=−\sqrt{x}\)

Exercise 1.5.42

Graph of a square root function.

Graph of a square root function.

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

Exercise 1.5.43

Graph of a parabola.

Graph of a parabola.

Solution

\(f(x)=−(x+1)^2+2\)

Exercise 1.5.44

Graph of a cubic function.

Graph of a cubic function.

Exercise 1.5.45

Graph of a square root function.

Graph of a square root function.

Solution

\(f(x)=\sqrt{−x}+1\)

Exercise 1.5.46

Graph of an absolute function.

Graph of an absolute function.

For the following exercises, determine whether the function is odd, even, or neither.

Exercise 1.5.47

\(f(x)=3x^4\)

Solution

even

Exercise 1.5.48

\(g(x)=\sqrt{x}\)

Exercise 1.5.49

\(h(x)=\dfrac{1}{x}+3x\)

Solution

odd

Exercise 1.5.50

\(f(x)=(x−2)^2\)

Exercise 1.5.51

\(g(x)=2x^4\)

Solution

even

Exercise 1.5.52

\(h(x)=2x−x^3\)

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\).

Exercise 1.5.53

\(g(x)=−f(x)\)

Solution

The graph of \(g\) is a vertical reflection (across the x-axis) of the graph of \(f\).

Exercise 1.5.54

\(g(x)=f(−x)\)

Exercise 1.5.55

\(g(x)=4f(x)\)

Solution

The graph of \(g\) is a vertical stretch by a factor of 4 of the graph of \(f\).

Exercise 1.5.56

\(g(x)=6f(x)\)

Exercise 1.5.57

\(g(x)=f(5x)\)

Solution

The graph of \(g\) is a horizontal compression by a factor of 15 of the graph of \(f\).

Exercise 1.5.58

\(g(x)=f(2x)\)

Exercise 1.5.59

\(g(x)=f(\frac{1}{3}x)\)

Solution

The graph of \(g\) is a horizontal stretch by a factor of 3 of the graph of \(f\).

Exercise 1.5.60

\(g(x)=f(\frac{1}{5}x)\)

Exercise 1.5.61

\(g(x)=3f(−x)\)

Solution

The graph of \(g\) is a horizontal reflection across the y-axis and a vertical stretch by a factor of 3 of the graph of \(f\).

Exercise 1.5.62

\(g(x)=−f(3x)\)

For the following exercises, write a formula for the function \(g\) that results when the graph of a given toolkit function is transformed as described.

Exercise 1.5.63

The graph of \(f(x)=|x|\) is reflected over the y-axis and horizontally compressed by a factor of \(\frac{1}{4}\) .

Solution

\(g(x)=|−4x|\)

Exercise 1.5.64

The graph of \(f(x)=\sqrt{x}\) is reflected over the x-axis and horizontally stretched by a factor of 2.

Exercise 1.5.65

The graph of \(f(x)=\dfrac{1}{x^2}\) is vertically compressed by a factor of \(\dfrac{1}{3}\), then shifted to the left 2 units and down 3 units.

Solution

\(g(x)=\dfrac{1}{3(x+2)^2}−3\)

Exercise 1.5.66

The graph of \(f(x)=\dfrac{1}{x}\) is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

Exercise 1.5.67

The graph of \(f(x)=x^2\) is vertically compressed by a factor of \(\frac{1}{2}\), then shifted to the right 5 units and up 1 unit.

\(g(x)=\dfrac{1}{2}(x-5)^2+1\)

Exercise 1.5.68

The graph of \(f(x)=x^2\) is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

Exercise 1.5.69

\(g(x)=4(x+1)^2−5\)

Solution

The graph of the function \(f(x)=x^2\) is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

Graph of a parabola.

Graph of a parabola.

Exercise 1.5.70

\(g(x)=5(x+3)^2−2\)

Exercise 1.5.71

\(h(x)=−2|x−4|+3\)

Solution

The graph of \(f(x)=|x|\) is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

Graph of an absolute value function.

Graph of an absolute value function.

Exercise 1.5.72

\(k(x)=−3\sqrt{x}−1\)

Exercise 1.5.73

\(m(x)=\dfrac{1}{2}x^3\)

Solution

The graph of the function \(f(x)=x^3\) is compressed vertically by a factor of \(\dfrac{1}{2}\).

Graph of a cubic function.

Graph of a cubic function.

Exercise 1.5.74

\(n(x)=\dfrac{1}{3}|x−2|\)

Exercise 1.5.75

\(p(x)=(\dfrac{1}{3}x)^3−3\)

Solution

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

Graph of a cubic function.

Graph of a cubic function.

Exercise 1.5.76

\(q(x)=(\dfrac{1}{4}x)^3+1\)

Exercise 1.5.77

\(a(x)=\sqrt{−x+4}\)

Solution

The graph of \(f(x)=\sqrt{x}\) is shifted right 4 units and then reflected across the vertical line \(x=4\).

Graph of a square root function.

Graph of a square root function.

For the following exercises, use the graph in Figure 1.5.30 to sketch the given transformations.

Figure 1.5.30: Graph of a polynomial.

Exercise 1.5.78

\(g(x)=f(x)−2\)

Exercise 1.5.79

\(g(x)=−f(x)\)

Solution

Graph of a polynomial.

Exercise 1.5.80

\(g(x)=f(x+1)\)

Exercise 1.5.81

\(g(x)=f(x−2)\)

Solution

Graph of a polynomial.

1.6: Absolute Value Functions

Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

Section Exercise

Verbal

Exercise 1.6.1

How do you solve an absolute value equation?

Solution

Isolate the absolute value term so that the equation is of the form \(|A|=B\). Form one equation by setting the expression inside the absolute value symbol, \(A\), equal to the expression on the other side of the equation, \(B\). Form a second equation by setting \(A\) equal to the opposite of the expression on the other side of the equation, \(−B\). Solve each equation for the variable.

Exercise 1.6.2

How can you tell whether an absolute value function has two x-intercepts without graphing the function?

Exercise 1.6.3

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

Solution

The graph of the absolute value function does not cross the x-axis, so the graph is either completely above or completely below the x-axis.

Exercise 1.6.4

How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

Exercise 1.6.5

How do you solve an absolute value inequality algebraically?

Solution

First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

Algebraic

Exercise 1.6.6

Describe all numbers \(x\) that are at a distance of 4 from the number 8. Express this using absolute value notation.

Exercise 1.6.7

Describe all numbers \(x\) that are at a distance of \(\dfrac{1}{2}\) from the number −4. Express this using absolute value notation.

Solution

\(|x+4|= \frac{1}{2}\)

Exercise 1.6.8

Describe the situation in which the distance that point \(x\) is from 10 is at least 15 units. Express this using absolute value notation.

Exercise 1.6.9

Find all function values \(f(x)\) such that the distance from \(f(x)\) to the value 8 is less than 0.03 units. Express this using absolute value notation.

Solution

\(|f(x)−8|<0.03\)

For the following exercises, solve the equations below and express the answer using set notation.

Exercise 1.6.10

\(|x+3|=9\)

Exercise 1.6.11

\(|6−x|=5\)

Solution

\({1,11}\)

Exercise 1.6.12

\(|5x−2|=11\)

Exercise 1.6.13

\(|4x−2|=11\)

Solution

\(\{\frac{9}{4}, \frac{13}{4}\}\)

Exercise 1.6.14

\(2|4−x|=7\)

Exercise 1.6.15

\(3|5−x|=5\)

Solution

\(\{\frac{10}{3},\frac{20}{3}\}\)

Exercise 1.6.16

\(3|x+1|−4=5\)

Exercise 1.6.17

\(5|x−4|−7=2\)

Solution

\(\{\frac{11}{5}, \frac{29}{5}\}\)

Exercise 1.6.18

\(0=−|x−3|+2\)

Exercise 1.6.19

\(2|x−3|+1=2\)

Solution

\(\{\frac{5}{2}, \frac{7}{2}\}\)

Exercise 1.6.20

\(|3x−2|=7\)

Exercise 1.6.21

\(|3x−2|=−7\)

Solution

No solution

Exercise 1.6.22

\(|\frac{1}{2}x−5|=11\)

Exercise 1.6.23

\(| \frac{1}{3}x+5|=14\)

Solution

\(\{−57,27\}\)

Exercise 1.6.24

\(−|\frac{1}{3}x+5|+14=0\)

For the following exercises, find the x- and y-intercepts of the graphs of each function.

Exercise 1.6.25

\(f(x)=2|x+1|−10\)

Solution

\((0,−8)\); \((−6,0)\), \((4,0)\)

Exercise 1.6.26

\(f(x)=4|x−3|+4\)

Exercise 1.6.27

\(f(x)=−3|x−2|−1\)

Solution

\((0,−7)\); no x-intercepts

Exercise 1.6.28

\(f(x)=−2|x+1|+6\)

For the following exercises, solve each inequality and write the solution in interval notation.

Exercise 1.6.29

\(| x−2 |>10\)

Solution

\((−\infty,−8)\cup(12,\infty)\)

Exercise 1.6.30

\(2|v−7|−4\geq42\)

Exercise 1.6.31

\(|3x−4|\geq8\)

Solution

\(−\dfrac{4}{3}{\leq}x\leq4\)

Exercise 1.6.32

\(|x−4|\geq8\)

Exercise 1.6.33

\(|3x−5|\geq-13\)

Solution

\((−\infty,− \frac{8}{3}]\cup\left[6,\infty\right)\)

Exercise 1.6.34

\(|3x−5|\geq−13\)

Exercise 1.6.35

\(|\frac{3}{4}x−5|\geq7\)

Solution

\((-\infty,-\frac{8}{3}]\cup\left[16,\infty\right)\)

Exercise 1.6.36

\(|\frac{3}{4}x−5|+1\leq16\)

Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

Exercise 1.6.37

\(y=|x−1|\)

Solution

Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).

Exercise 1.6.38

\(y=|x+1|\)

Exercise 1.6.39

\(y=|x|+1\)

Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3)

For the following exercises, graph the given functions by hand.

Exercise 1.6.40

\(y=|x|−2\)

Exercise 1.6.41

\(y=−|x|\)

Solution

Graph of an absolute function.

Exercise 1.6.42

\(y=−|x|−2\)

Exercise 1.6.43

\(y=−|x−3|−2\)

Solution

Graph of an absolute function.

Exercise 1.6.44

\(f(x)=−|x−1|−2\)

Exercise 1.6.45

\(f(x)=−|x+3|+4\)

Solution

Graph of an absolute function.

Exercise 1.6.46

\(f(x)=2|x+3|+1\)

Exercise

\(f(x)=3|x−2|+3\)

Solution

Graph of an absolute function.

Exercise 1.6.47

\(f(x)=|2x−4|−3\)

Exercise 1.6.48

\(f(x)=|3x+9|+2\)

Solution

Graph of an absolute function.

Exercise 1.6.49

\(f(x)=−|x−1|−3\)

Exercise 1.6.50

\(f(x)=−|x+4|−3\)

Solution

Graph of an absolute function.

Exercise 1.6.51

\(f(x)=\frac{1}{2}|x+4|−3\)

Technology

Exercise 1.6.52

Use a graphing utility to graph \(f(x)=10|x−2|\) on the viewing window \([0,4]\). Identify the corresponding range. Show the graph.

Solution

range: \([0,20]\)

Graph of an absolute value function.

Exercise 1.6.53

Use a graphing utility to graph \(f(x)=−100|x|+100\) on the viewing window \([−5,5]\). Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

Exercise 1.6.54

\(f(x)=−0.1|0.1(0.2−x)|+0.3\)

Solution

x-intercepts:

Graph of an absolute function.

Exercise 1.6.55

\(f(x)=4 \times10^{9}|x−(5 \times 10^9)|+2 \times10^9\)

Extensions

For the following exercises, solve the inequality.

Exercise 1.6.56

\(|−2x− \frac{2}{3}(x+1)|+3>−1\)

Solution

\((−\infty,\infty)\)

Exercise 1.6.57

If possible, find all values of a such that there are no x-intercepts for \(f(x)=2|x+1|+a\).

Exercise 1.6.58

If possible, find all values of a such that there are no y-intercepts for \(f(x)=2|x+1|+a\).

Solution

There is no solution for a that will keep the function from having a y-intercept. The absolute value function always crosses the y-intercept when \(x=0\).

Real-World Applications

Exercise 1.6.59

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and \(x\) represents the distance from city B to city A, express this using absolute value notation.

Exercise 1.6.60

The true proportion \(p\) of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Solution

\(|p−0.08|\leq0.015\)

Exercise 1.6.61

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable \(x\) for the score.

Exercise 1.6.62

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using \(x\) as the diameter of the bearing, write this statement using absolute value notation.

Solution

\(|x−5.0|\leq0.01\)

Exercise 1.6.63

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is \(x\) inches, express the tolerance using absolute value notation.

1.7: Inverse Functions

If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. In this section, we will consider the reverse nature of functions.

Section Exercises

Verbal

Exercise 1.7.1

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Solution

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that \(y\)-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no \(y\)-values repeat and the function is one-to-one.

Exercise 1.7.2

Why do we restrict the domain of the function \(f(x)=x^2\) to find the function’s inverse?

Exercise 1.7.3

Can a function be its own inverse? Explain.

Solution

Yes. For example, \(f(x)=\frac{1}{x}\) is its own inverse.

Exercise 1.7.4

Are one-to-one functions either always increasing or always decreasing? Why or why not?

Exercise 1.7.5

How do you find the inverse of a function algebraically?

Solution

Given a function \(y=f(x)\), solve for \(x\) in terms of \(y\). Interchange the \(x\) and \(y\). Solve the new equation for \(y\). The expression for \(y\) is the inverse, \(y=f^{-1}(x)\).

Algebraic

Exercise 1.7.6

Show that the function \(f(x)=a−x\) is its own inverse for all real numbers \(a\).

For the following exercises, find \(f^{-1}(x)\) for each function.

Exercise 1.7.7

\(f(x)=x+3\)

Solution

\(f^{-1}(x)=x−3\)

Exercise 1.7.8

\(f(x)=x+5\)

Exercise 1.7.9

\(f(x)=2−x\)

Solution

\(f^{-1}(x)=2−x\)

Exercise 1.7.10:

\(f(x)=3−x\)

Exercise 1.7.11

\(f(x)=\frac{x}{x+2}\)

Solution

\(f^{-1}(x)=\frac{−2x}{x−1}\)

Exercise 1.7.12

\(f(x)=\frac{2x+3}{5x+4}\)

For the following exercises, find a domain on which each function \(f\) is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain.

Exercise 1.7.13

\(f(x)=(x+7)^2\)

Solution

domain of \(f(x)\): \(\left[−7,\infty\right)\); \(f^{-1}(x)=\sqrt{x}−7\)

Exercise 1.7.14

\(f(x)=(x−6)^2\)

Exercise 1.7.15

\(f(x)=x^2−5\)

Solution

domain of \(f(x)\): \(\left[0,\infty\right)\); \(f^{-1}(x)=\sqrt{x+5}\)

Exercise 1.7.16

Given \(f(x)=\frac{x}{2+x}\) and \(g(x)=\frac{2x}{1-x}:\)

a. Find \(f(g(x))\) and \(g(f(x))\).

b. What does the answer tell us about the relationship between \(f(x)\) and \(g(x)?\)

Solution

a. \(f(g(x))=x\) and \(g(f(x))=x\). b. This tells us that \(f\) and \(g\) are inverse functions

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions.

Exercise 1.7.17

\(f(x)=\sqrt[3]{x-1}\) and \(g(x)=x^3+1\)

Solution

 \(f(g(x))=x\), \(g(f(x))=x\)

Exercise 1.7.18

\(f(x)=−3x+5\) and \(g(x)=\frac{x-5}{-3}\)

Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

Exercise 1.7.19

\(f(x)=\sqrt{x}\)

Solution

one-to-one

Exercise 1.7.20

\(f(x)=\sqrt[3]{3x+1}\)

Exercise 1.7.21

\(f(x)=−5x+1\)

Solution

one-to-one

Exercise 1.7.22

\(f(x)=x^3−27\)

For the following exercises, determine whether the graph represents a one-to-one function.

Exercise 1.7.23

Graph of a parabola.

Solution

not one-to-one

Exercise 1.7.24

Graph of a step-function.

For the following exercises, use the graph of \(f\) shown in Figure 1.7.11.

Graph of a line.

Figure 1.7.11: Graph of a line

Exercise 1.7.25

Find \(f(0)\).

Solution

3

Exercise 1.7.26

Solve \(f(x)=0\).

Exercise 1.7.27

Find \(f^{-1}(0)\).

Solution

2

Exercise 1.7.28

Solve \(f^{-1}(x)=0\).

For the following exercises, use the graph of the one-to-one function shown in Figure 1.7.12.

Graph of a square root function.
Figure 1.7.12: Graph of a square root function.

Exercise 1.7.29

Sketch the graph of \(f^{-1}\).

Solution

Graph of a square root function and its inverse.

Exercise 1.7.30

Find \(f(6)\) and \(f^{-1}(2)\).

Exercise 1.7.31

If the complete graph of \(f\) is shown, find the domain of \(f\).

Solution

\([2,10]\)

Exercise 1.7.32

If the complete graph of \(f\) is shown, find the range of \(f\)

Numeric

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one.

Exercise 1.7.33

If \(f(6)=7\), find \(f^{-1}(7)\).

Solution

6

Exercise 1.7.34

If \(f(3)=2\), find \(f^{-1}(2)\).

Exercise 1.7.35

If \(f^{-1}(−4)=−8\), find \(f(−8)\).

Solution

-4

Exercise 1.7.36

If \(f^{-1}(−2)\)=−1, find \(f(−1)\).

For the following exercises, use the values listed in Table 1.7.6 to evaluate or solve.

\(x\) \(f(x)\)
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1

Table 1.7.6

Exercise 1.7.37

Find \(f(1)\).

Solution

0

Exercise 1.7.38

Solve \(f(x)=3\).

Exercise 1.7.39

Find \(f^{-1}(0)\).

Solution

1

Exercise 1.7.40

Solve \(f^{-1}(x)=7\).

Exercise 1.7.41

Use the tabular representation of \(f\) in Table 1.7.7 to create a table for f^{-1}(x).

\(x\)

3 6 9 13 14

\(f(x)\)

1 4 7 12 16

Table 1.7.7

Solution

\(x\)

1 4 7 12 16

\(f^{-1}(x)\)

3 6 9 13 14

Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

Exercise 1.7.42

\(f(x)=\dfrac{3}{x-2}\)

Exercise 1.7.43

\(f(x)=x^3−1\)

Solution

\(f^{-1}(x)=(1+x)^{1/3}\)

Graph of a cubic function and its inverse.

Exercise 1.7.44

Find the inverse function of \(f(x)=\frac{1}{x-1}\). Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

Real-World Applications

Exercise 1.7.45

To convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit, we use the formula \(f(x)=\frac{9}{5}x+32\). Find the inverse function, if it exists, and explain its meaning.

Solution

\(f^{-1}(x)=\frac{5}{9}(x−32)\). Given the Fahrenheit temperature, \(x\), this formula allows you to calculate the Celsius temperature.

Exercise 1.7.46

The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2{\pi}r\). Express the radius of a circle as a function of its circumference. Call this function \(r(C)\). Find \(r(36\pi)\) and interpret its meaning.

Exercise 1.7.47

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, \(t\), in hours given by \(d(t)=50t\). Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function \(t(d)\). Find \(t(180)\) and interpret its meaning.

Solution

\(t(d)=\frac{d}{50}\), \(t(180)=\frac{180}{50}\). The time for the car to travel 180 miles is 3.6 hours.