1.R: Functions (Review)

1.1: Functions and Function Notation

For the exercises 1-4, determine whether the relation is a function.

1) \(\{(a,b),(c,d),(e,d)\}\)

Answer

function

2) \(\{(5,2),(6,1),(6,2),(4,8)\}\)

3) \(y^2+4=x\), for \(x\) the independent variable and \(y\) the dependent variable

Answer

not a function

4) Is the graph in the Figure below a function?

For the exercises 5-6, evaluate the function at the indicated values: \(f(-3); f(2); f(-a); -f(a); f(a+h)\)

5) \(f(x)=-2x^2+3x\)

Answer

\(f(-3)=-27; f(2)=-2;f(-a)=-2a^2-3a;-f(a)=2a^2-3a;f(a+h)=-2a^2+3a-4ah+3h-2h^2\)

6) \(f(x)=2|3x-1|\)

For the exercises 7-8, determine whether the functions are one-to-one.

7) \(f(x)=-3 x+5\)

Answer

one-to-one

8) \(f(x)=|x-3|\)

For the exercises 9-11, use the vertical line test to determine if the relation whose graph is provided is a function.

9)

Answer

function

10)

11)

Answer

function

For the exercises 12-13, graph the functions.

12) \(f(x)=|x+1|\)

13) \(f(x)=x^{2}-2\)

Answer

For the exercises 14-17, use the Figure below to approximate the values.

14) \(f(2)\)

15) \(f(-2)\)

Answer

\(2\)

16) If \(f(x)=-2\), then solve for \(x\)

17) If \(f(x)=1\), then solve for \(x\)

Answer

\(x=-1.8\) or \(x=1.8\)

For the exercises 18-19, use the function \(h(t)=-16 t^{2}+80t\) to find the values.

18) \(\dfrac{h(2)-h(1)}{2-1}\)

19) \(\dfrac{h(a)-h(1)}{a-1}\)

Answer

\(\dfrac{-64+80 a-16 a^{2}}{-1+a}=-16 a+64\)

1.2: Domain and Range

For the exercises 1-4, find the domain of each function, expressing answers using interval notation.

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

2) \(f(x)=\frac{x-3}{x^{2}-4 x-12}\)

Answer

\((-\infty,-2) \cup(-2,6) \cup(6, \infty)\)

3)

4) Graph this piecewise function: \(f(x)=\left\{\begin{array}{ll}{x+1} & {x<-2} \\ {-2 x-3} & {x \geq-2}\end{array}\right.\)

Answer

1.3: Rates of Change and Behavior of Graphs

For the exercises 1-3, find the average rate of change of the functions from \(x=1\) to \(x=2\)

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

2) \(f(x)=10 x^{2}+x\)

Answer

\(31\)

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

For the exercises 4-6, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

4)

Answer

increasing \((2, \infty)\); decreasing \((-\infty, 2)\)

5)

6)

Answer

increasing \((-3,1)\); constant \((-\infty,-3) \cup(1, \infty)\)

7) Find the local minimum of the function graphed in Exercise 4.

8) Find the local extrema for the function graphed in Exercise 5.

Answer

local minimum \((-2,-3)\); local maximum \((1,3)\)

9) For the graph in the Figure in Exercise 10, the domain of the function is \([-3,3]\). The range is \([-10,10]\). Find the absolute minimum of the function on this interval.

10) Find the absolute maximum of the function graphed in the Figure below.

Answer

\((-1.8,10)\)

1.4: Composition of Functions

For the exercises 1-5, find \((f \circ g)(x)\) and \((g \circ f)(x)\) for each pair of functions.

1) \(f(x)=4-x, g(x)=-4x\)

2) \(f(x)=3 x+2, g(x)=5-6x\)

Answer

\((f \circ g)(x)=17-18 x ;(g \circ f)(x)=-7-18x\)

3) \(f(x)=x^{2}+2 x, g(x)=5 x+1\)

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

Answer

\((f \circ g)(x)=\sqrt{\dfrac{1}{x}+2} ;(g \circ f)(x)=\dfrac{1}{\sqrt{x+2}}\)

5) \(f(x)=\dfrac{x+3}{2}, g(x)=\sqrt{1-x}\)

For the exercises 6-9, find \((f \circ g)\) and the domain for \((f \circ g)(x)\) for each pair of functions.

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

Answer

\((f \circ g)(x)=\dfrac{1+x}{1+4 x}, x \neq 0, x \neq-\dfrac{1}{4}\)

7) \(f(x)=\dfrac{1}{x+3}, g(x)=\dfrac{1}{x-9}\)

8) \(f(x)=\dfrac{1}{x}, g(x)=\sqrt{x}\)

Answer

\((f \circ g)(x)=\frac{1}{\sqrt{x}}, x>0\)

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

For the exercises 10-11, express each function \(H\) as a composition of two functions \(f\) and \(g\) where \(H(x)=(f \circ g)(x)\)

10) \(H(x)=\sqrt{\frac{2 x-1}{3 x+4}}\)

Answer

sample: \(g(x)=\dfrac{2 x-1}{3 x+4}; f(x)=\sqrt{x}\)

11) \(H(x)=\dfrac{1}{\left(3 x^{2}-4\right)^{-3}}\)

1.5: Transformation of Functions

For the exercises 1-8, sketch a graph of the given function.

1) \(f(x)=(x-3)^{2}\)

Answer

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

3) \(f(x)=\sqrt{x}+5\)

Answer

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

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

Answer

6) \(f(x)=5 \sqrt{-x}-4\)

7) \(f(x)=4[|x-2|-6]\)

Answer

8) \(f(x)=-(x+2)^{2}-1\)

For the exercises 9-10, sketch the graph of the function \(g\) if the graph of the function \(f\) is shown in the Figure below.

9) \(g(x)=f(x-1)\)

Answer

10) \(g(x)=3 f(x)\)

For the exercises 11-12, write the equation for the standard function represented by each of the graphs below.

11)

Answer

\(f(x)=|x-3|\)

12)

For the exercises 13-15, determine whether each function below is even, odd, or neither.

13) \(f(x)=3 x^{4}\)

Answer

even

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

15) \(h(x)=\frac{1}{x}+3 x\)

Answer

odd

For the exercises 16-18, analyze the graph and determine whether the graphed function is even, odd, or neither.

16)

17)

Answer

even

18)

1.6: Absolute Value Functions

For the exercises 1-3, write an equation for the transformation of \(f(x)=|x|\).

1)

Answer

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

2)

3)

Answer

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

For the exercises 4-6, graph the absolute value function.

4) \(f(x)=|x-5|\)

5) \(f(x)=-|x-3|\)

Answer

6) \(f(x)=|2 x-4|\)

For the exercises 7-8, solve the absolute value equation.

7) \(|x+4|=18\)

Answer

\(x=-22, x=14\)

8) \(\left|\dfrac{1}{3} x+5\right|=\left|\dfrac{3}{4} x-2\right|\)

For the exercises 9-10, solve the inequality and express the solution using interval notation.

9) \(|3 x-2|<7\)

Answer

\(\left(-\dfrac{5}{3}, 3\right)\)

10) \(\left|\dfrac{1}{3} x-2\right| \leq 7\)

1.7: Inverse Functions

For the exercises 1-2, find \(f^{-1}(x)\) for each function.

1) \(f(x)=9+10 x\)

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

Answer

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

3) For the following exercise, find a domain on which the 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. \[f(x)=x^{2}+1\]

4) Given \(f(x)=x^{3}-5\) and \(g(x)=\sqrt[3]{x+5} \) :

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

Answer

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

For the exercises 5-8, use a graphing utility to determine whether each function is one-to-one.

5) \(f(x)=\dfrac{1}{x}\)

Answer

The function is one-to-one.

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

Answer

The function is not one-to-one.

7) If \(f(5)=2,\) find \(f^{-1}(2)\)

Answer

\(5\)

8) If \(f(1)=4,\) find \(f^{-1}(4)\)

Practice Test

For the exercises 1-2, determine whether each of the following relations is a function.

1) \(y=2 x+8\)

Answer

The relation is a function.

2) \(\{(2,1),(3,2),(-1,1),(0,-2)\}\)

For the exercises 3-4, evaluate the function \(f(x)=-3 x^{2}+2 x\) at the given input.

3) \(f(-2)\)

Answer

\(-16\)

4) \(f(a)\)

5) Show that the function \(f(x)=-2(x-1)^{2}+3\) is not one-to-one.

Answer

The graph is a parabola and the graph fails the horizontal line test.

6) Write the domain of the function \(f(x)=\sqrt{3-x}\) in interval notation.

7) Given \(f(x)=2 x^{2}-5 x,\) find \(f(a+1)-f(1)\)

Answer

\(2 a^{2}-a\)

8) Graph the function \(f(x)=\left\{\begin{array}{ccc}{x+1} & {\text { if }} & {-2<x<3} \\ {-x} & {\text { if }} & {x \geq 3}\end{array}\right.\)

9) Find the average rate of change of the function \(f(x)=3-2 x^{2}+x\) by finding \(\dfrac{f(b)-f(a)}{b-a}\)

Answer

\(-2(a+b)+1\)

For the exercises 10-11, use the functions \(f(x)=3-2 x^{2}+x\) and \(g(x)=\sqrt{x}\) to find the composite functions.

10) \((g \circ f)(x)\)

11) \((g \circ f)(1)\)

Answer

\(\sqrt{2}\)

12) Express \(H(x)=\sqrt[3]{5 x^{2}-3 x}\) a composition of two functions, \(f\) and \(g,\) where \((f \circ g)(x)=H(x)\)

For the exercises 13-14, graph the functions by translating, stretching, and/or compressing a toolkit function.

13) \(f(x)=\sqrt{x+6}-1\)

Answer

14) \(f(x)=\dfrac{1}{x+2}-1\)

For the exercises 15-17, determine whether the functions are even, odd, or neither.

15) \(f(x)=-\dfrac{5}{x^{2}}+9 x^{6}\)

Answer

even

16) \(f(x)=-\dfrac{5}{x^{3}}+9 x^{5}\)

17) \(f(x)=\dfrac{1}{x}\)

Answer

odd

18) Graph the absolute value function \(f(x)=-2|x-1|+3\).

19) Solve \(|2 x-3|=17\).

Answer

\(x=-7\) and \(x=10\)

20) Solve \(-\left|\dfrac{1}{3} x-3\right| \geq 17\). Express the solution in interval notation.

For the exercises 21-22, find the inverse of the function.

21) \(f(x)=3 x-5\)

Answer

\(f^{-1}(x)=\dfrac{x+5}{3}\)

22) \(f(x)=\dfrac{4}{x+7}\)

For the exercises 23-26, use the graph of \(g\) shown in the Figure below.

23) On what intervals is the function increasing?

Answer

\((-\infty,-1.1)\) and \((1.1, \infty)\)

24) On what intervals is the function decreasing?

25) Approximate the local minimum of the function. Express the answer as an ordered pair.

Answer

\((1.1,-0.9)\)

26) Approximate the local maximum of the function. Express the answer as an ordered pair.

For the exercises 27-29, use the graph of the piecewise function shown in the Figure below.

27) Find \(f(2)\).

Answer

\(f(2)=2\)

28) Find \(f(-2)\).

29) Write an equation for the piecewise function.

Answer

\(f(x)=\left\{\begin{array}{cl}{|x|} & {\text { if } x \leq 2} \\ {3} & {\text { if } x>2}\end{array}\right.\)

For the exercises 30-35, use the values listed in the Table below.

\(x\)	\(F(x)\)
0	1
1	3
2	5
3	7
4	9
5	11
6	13
7	15
8	17

30) Find \(F(6)\).

31) Solve the equation \(F(x)=5\)

Answer

\(x=2\)

32) Is the graph increasing or decreasing on its domain?

33) Is the function represented by the graph one-to-one?

Answer

yes

34) Find \(F^{-1}(15)\).

35) Given \(f(x)=-2 x+11,\) find \(f^{-1}(x)\).

Answer

\(f^{-1}(x)=-\dfrac{x-11}{2}\)

Contributors and Attributions

• Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.