 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?

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 . 10 . 11 . 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.

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:

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from

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 . 28 . 29 . 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.

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.

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 . 56 . 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 . 61 . 62 . Absolute Value Functions

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

63 . 64 . 65 . 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 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 $f −1 (4). f −1 (4).$