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.