Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra and Trigonometry 2e

Review Exercises

Algebra and Trigonometry 2eReview Exercises

Review Exercises

Functions and Function Notation

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

1.

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

2.

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

3.

y 2 +4=x, y 2 +4=x, for x x the independent variable and y y the dependent variable

4.

Is the graph in Figure 1 a function?

Graph of a parabola.
Figure 1

For the following exercises, evaluate f(−3);f(2);f(−a);−f(a);f(a+h). f(−3);f(2);f(−a);−f(a);f(a+h).

5.

f(x)=−2 x 2 +3x f(x)=−2 x 2 +3x

6.

f(x)=2| 3x−1 | f(x)=2| 3x−1 |

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

7.

f(x)=−3x+5 f(x)=−3x+5

8.

f(x)=| x−3 | f(x)=| x−3 |

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

9.
Graph of a cubic function.
10.
Graph of a relation.
11.
Graph of a relation.

For the following exercises, graph the functions.

12.

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

13.

f(x)= x 2 −2 f(x)= x 2 −2

For the following exercises, use Figure 2 to approximate the values.

Graph of a parabola.
Figure 2
14.

f(2) f(2)

15.

f(−2) f(−2)

16.

If f(x)=−2, f(x)=−2, then solve for x. x.

17.

If f(x)=1, f(x)=1, then solve for x. x.

For the following exercises, use the function h(t)=−16 t 2 +80t h(t)=−16 t 2 +80t to find the values in simplest form.

18.

h(2)−h(1) 2−1 h(2)−h(1) 2−1

19.

h(a)−h(1) a−1 h(a)−h(1) a−1

Domain and Range

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

20.

f(x)= 2 3x+2 f(x)= 2 3x+2

21.

f(x)= x−3 x 2 −4x−12 f(x)= x−3 x 2 −4x−12

22.

f(x)= x−6 x−4 f(x)= x−6 x−4

23.

Graph this piecewise function: f(x)={ x+1       x<−2 −2x−3  x≥−2 f(x)={ x+1       x<−2 −2x−3  x≥−2

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from x=1to x=2. x=1to x=2.

24.

f(x)=4x−3 f(x)=4x−3

25.

f(x)=10 x 2 +x f(x)=10 x 2 +x

26.

f(x)=− 2 x 2 f(x)=− 2 x 2

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

27.
Graph of a parabola.
28.
Graph of a cubic function.
29.
Graph of a function.
30.

Find the local minimum of the function graphed in Exercise 3.27.

31.

Find the local extrema for the function graphed in Exercise 3.28.

32.

For the graph in Figure 3, the domain of the function is [ −3,3 ]. [ −3,3 ]. The range is [ −10,10 ]. [ −10,10 ]. Find the absolute minimum of the function on this interval.

33.

Find the absolute maximum of the function graphed in Figure 3.

Graph of a cubic function.
Figure 3

Composition of Functions

For the following exercises, find (f∘g)(x) (f∘g)(x) and (g∘f)(x) (g∘f)(x) for each pair of functions.

34.

f(x)=4−x,g(x)=−4x f(x)=4−x,g(x)=−4x

35.

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

36.

f(x)= x 2 +2x,g(x)=5x+1 f(x)= x 2 +2x,g(x)=5x+1

37.

f(x)= x+2 ,g(x)= 1 x f(x)= x+2 ,g(x)= 1 x

38.

f(x)= x+3 2 ,g(x)= 1−x f(x)= x+3 2 ,g(x)= 1−x

For the following exercises, find ( f∘g ) ( f∘g ) and the domain for ( f∘g )(x) ( f∘g )(x) for each pair of functions.

39.

f(x)= x+1 x+4 ,g(x)= 1 x f(x)= x+1 x+4 ,g(x)= 1 x

40.

f(x)= 1 x+3 ,g(x)= 1 x−9 f(x)= 1 x+3 ,g(x)= 1 x−9

41.

f(x)= 1 x ,g(x)= x f(x)= 1 x ,g(x)= x

42.

f(x)= 1 x 2 −1 ,g(x)= x+1 f(x)= 1 x 2 −1 ,g(x)= x+1

For the following exercises, express each function H H as a composition of two functions f f and g g where H(x)=(f∘g)(x). H(x)=(f∘g)(x).

43.

H(x)= 2x−1 3x+4 H(x)= 2x−1 3x+4

44.

H(x)= 1 (3 x 2 −4) −3 H(x)= 1 (3 x 2 −4) −3

Transformation of Functions

For the following exercises, sketch a graph of the given function.

45.

f(x)= (x−3) 2 f(x)= (x−3) 2

46.

f(x)= (x+4) 3 f(x)= (x+4) 3

47.

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

48.

f(x)=− x 3 f(x)=− x 3

49.

f(x)= −x 3 f(x)= −x 3

50.

f(x)=5 −x −4 f(x)=5 −x −4

51.

f(x)=4[ | x−2 |−6 ] f(x)=4[ | x−2 |−6 ]

52.

f(x)=− (x+2) 2 −1 f(x)=− (x+2) 2 −1

For the following exercises, sketch the graph of the function g g if the graph of the function f f is shown in Figure 4.

Graph of f(x)
Figure 4
53.

g(x)=f(x−1) g(x)=f(x−1)

54.

g(x)=3f(x) g(x)=3f(x)

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

55.
Graph of an absolute function.
56.
Graph of a half circle.

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

57.

f(x)=3 x 4 f(x)=3 x 4

58.

g(x)= x g(x)= x

59.

h(x)= 1 x +3x h(x)= 1 x +3x

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

60.
Graph of a parabola.
61.
Graph of a parabola.
62.
Graph of a cubic function.

Absolute Value Functions

For the following exercises, write an equation for the transformation of f(x)=| x |. f(x)=| x |.

63.
Graph of f(x).
64.
Graph of f(x).
65.
Graph of f(x).

For the following exercises, graph the absolute value function.

66.

f(x)=| x−5 | f(x)=| x−5 |

67.

f(x)=−| x−3 | f(x)=−| x−3 |

68.

f(x)=| 2x−4 | f(x)=| 2x−4 |

Inverse Functions

For the following exercises, find f −1 (x) f −1 (x) for each function.

69.

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

70.

f(x)= x x+2 f(x)= x x+2

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

71.

f(x)= x 2 +1 f(x)= x 2 +1

72.

Given f( x )= x 3 −5 f( x )= x 3 −5 and g(x)= x+5 3 : g(x)= x+5 3 :

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

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

73.

f(x)= 1 x f(x)= 1 x

74.

f(x)=−3 x 2 +x f(x)=−3 x 2 +x

75.

If f( 5 )=2, f( 5 )=2, find f −1 (2). f −1 (2).

76.

If f( 1 )=4, f( 1 )=4, find f −1 (4). f −1 (4).

PreviousNext
Order a print copy
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
Citation information

© Jun 28, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.