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# 10.2E: Exercises

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• OpenStax
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### Practice Makes Perfect

##### Exercise $$\PageIndex{19}$$ Find and Evaluate Composite Functions

In the following exercises, find

1. $$(f \circ g)(x)$$
2. $$(g \circ f)(x)$$
3. $$(f \cdot g)(x)$$
1. $$f(x)=4 x+3$$ and $$g(x)=2 x+5$$
2. $$f(x)=3 x-1$$ and $$g(x)=5 x-3$$
3. $$f(x)=6 x-5$$ and $$g(x)=4 x+1$$
4. $$f(x)=2 x+7$$ and $$g(x)=3 x-4$$
5. $$f(x)=3 x$$ and $$g(x)=2 x^{2}-3 x$$
6. $$f(x)=2 x$$ and $$g(x)=3 x^{2}-1$$
7. $$f(x)=2 x-1$$ and $$g(x)=x^{2}+2$$
8. $$f(x)=4 x+3$$ and $$g(x)=x^{2}-4$$

1.

1. $$8x+23$$
2. $$8x+11$$
3. $$8 x^{2}+26 x+15$$

3.

1. $$24x+1$$
2. $$24x-19$$
3. $$24x^{2}+19x-5$$

5.

1. $$6 x^{2}-9 x$$
2. $$18 x^{2}-9 x$$
3. $$6 x^{3}-9 x^{2}$$

7.

1. $$2 x^{2}+3$$
2. $$4 x^{2}-4 x+3$$
3. $$2 x^{3}-x^{2}+4 x-2$$
##### Exercise $$\PageIndex{20}$$ Find and Evaluate Composite Functions

In the following exercises, find the values described.

1. For functions $$f(x)=2 x^{2}+3$$ and $$g(x)=5x-1$$, find
1. $$(f \circ g)(-2)$$
2. $$(g \circ f)(-3)$$
3. $$(f \circ f)(-1)$$
2. For functions $$f(x)=5 x^{2}-1$$ and $$g(x)=4x−1$$, find
1. $$(f \circ g)(1)$$
2. $$(g \circ f)(-1)$$
3. $$(f \circ f)(2)$$
3. For functions $$f(x)=2x^{3}$$ and $$g(x)=3x^{2}+2$$, find
1. $$(f \circ g)(-1)$$
2. $$(g \circ f)(1)$$
3. $$(g \circ g)(1)$$
4. For functions $$f(x)=3 x^{3}+1$$ and $$g(x)=2 x^{2}=3$$, find
1. $$(f \circ g)(-2)$$
2. $$(g \circ f)(-1)$$
3. $$(g \circ g)(1)$$

1.

1. $$245$$
2. $$104$$
3. $$53$$

3.

1. $$250$$
2. $$14$$
3. $$77$$
##### Exercise $$\PageIndex{21}$$ Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

1. $$\begin{array}{l}{\{(-3,9),(-2,4),(-1,1),(0,0)}, {(1,1),(2,4),(3,9) \}}\end{array}$$
2. $$\begin{array}{l}{\{(9,-3),(4,-2),(1,-1),(0,0)}, {(1,1),(4,2),(9,3) \}}\end{array}$$
3. $$\begin{array}{l}{\{(-3,-5),(-2,-3),(-1,-1)}, {(0,1),(1,3),(2,5),(3,7) \}}\end{array}$$
4. $$\begin{array}{l}{\{(5,3),(4,2),(3,1),(2,0)}, {(1,-1),(0,-2),(-1,-3) \}}\end{array}$$

1. Function; not one-to-one

3. One-to-one function

##### Exercise $$\PageIndex{22}$$ Determine Whether a Function is One-to-One

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

1.

1. Figure 10.1.65

2. Figure 10.1.66

2.

1. Figure 10.1.67

2. Figure 10.1.68

3.

1. Figure 10.1.69

2. Figure 10.1.70

4.

1. Figure 10.1.71

2. Figure 10.1.72

1.

1. Not a function
2. Function; not one-to-one

3.

1. One-to-one function
2. Function; not one-to-one
##### Exercise $$\PageIndex{23}$$ Determine Whether a Function is One-to-One

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

1. $$\{(2,1),(4,2),(6,3),(8,4)\}$$
2. $$\{(6,2),(9,5),(12,8),(15,11)\}$$
3. $$\{(0,-2),(1,3),(2,7),(3,12)\}$$
4. $$\{(0,0),(1,1),(2,4),(3,9)\}$$
5. $$\{(-2,-3),(-1,-1),(0,1),(1,3)\}$$
6. $$\{(5,3),(4,2),(3,1),(2,0)\}$$

1. $$\begin{array}{l}{\text { Inverse function: }\{(1,2),(2,4),(3,6),(4,8)\} . \text { Domain: }\{1,2,3,4\} . \text { Range: }} {\{2,4,6,8\} .}\end{array}$$

3. $$\begin{array}{l}{\text { Inverse function: }\{(-2,0),(3,1),(7,2),(12,3)\} . \text { Domain: }\{-2,3,7,12\} \text { . }} {\text { Range: }\{0,1,2,3\}}\end{array}$$

5. $$\begin{array}{l}{\text { Inverse function: }\{(-3,-2),(-1,-1),(1,0),(3,1)\} . \text { Domain: }} {\{-3,-1,1,3\} . \text { Range: }\{-2,-1,0,1\}}\end{array}$$

##### Exercise $$\PageIndex{24}$$ Determine Whether a Function is One-to-One

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

1. Figure 10.1.73

2. Figure 10.1.74

3. Figure 10.1.75

4. Figure 10.1.76

1.

3.

##### Exercise $$\PageIndex{25}$$ Determine Whether the given functions are inverses

In the following exercises, determine whether or not the given functions are inverses.

1. $$f(x)=x+8$$ and $$g(x)=x-8$$
2. $$f(x)=x-9$$ and $$g(x)=x+9$$
3. $$f(x)=7 x$$ and $$g(x)=\frac{x}{7}$$
4. $$f(x)=\frac{x}{11}$$ and $$g(x)=11 x$$
5. $$f(x)=7 x+3$$ and $$g(x)=\frac{x-3}{7}$$
6. $$f(x)=5 x-4$$ and $$g(x)=\frac{x-4}{5}$$
7. $$f(x)=\sqrt{x+2}$$ and $$g(x)=x^{2}-2$$
8. $$f(x)=\sqrt{x-4}$$ and $$g(x)=x^{3}+4$$

1. $$g(f(x))=x,$$ and $$f(g(x))=x,$$ so they are inverses.

3. $$g(f(x))=x,$$ and $$f(g(x))=x,$$ so they are inverses.

5. $$g(f(x))=x,$$ and $$f(g(x))=x,$$ so they are inverses.

7. $$g(f(x))=x,$$ and $$f(g(x))=x,$$ so they are inverses (for nonnegative $$x )$$

##### Exercise $$\PageIndex{26}$$ Determine the inverse of a function

In the following exercises, find the inverse of each function.

1. $$f(x)=x-12$$
2. $$f(x)=x+17$$
3. $$f(x)=9 x$$
4. $$f(x)=8 x$$
5. $$f(x)=\frac{x}{6}$$
6. $$f(x)=\frac{x}{4}$$
7. $$f(x)=6 x-7$$
8. $$f(x)=7 x-1$$
9. $$f(x)=-2 x+5$$
10. $$f(x)=-5 x-4$$
11. $$f(x)=x^{2}+6, x \geq 0$$
12. $$f(x)=x^{2}-9, x \geq 0$$
13. $$f(x)=x^{3}-4$$
14. $$f(x)=x^{3}+6$$
15. $$f(x)=\frac{1}{x+2}$$
16. $$f(x)=\frac{1}{x-6}$$
17. $$f(x)=\sqrt{x-2}, x \geq 2$$
18. $$f(x)=\sqrt{x+8}, x \geq-8$$
19. $$f(x)=\sqrt{x-3}$$
20. $$f(x)=\sqrt{x+5}$$
21. $$f(x)=\sqrt{9 x-5}, x \geq \frac{5}{9}$$
22. $$f(x)=\sqrt{8 x-3}, x \geq \frac{3}{8}$$
23. $$f(x)=\sqrt{-3 x+5}$$
24. $$f(x)=\sqrt{-4 x-3}$$

1. $$f^{-1}(x)=x+12$$

3. $$f^{-1}(x)=\frac{x}{9}$$

5. $$f^{-1}(x)=6 x$$

7. $$f^{-1}(x)=\frac{x+7}{6}$$

9. $$f^{-1}(x)=\frac{x-5}{-2}$$

11. $$f^{-1}(x)=\sqrt{x-6}$$

13. $$f^{-1}(x)=\sqrt{x+4}$$

15. $$f^{-1}(x)=\frac{1}{x}-2$$

17. $$f^{-1}(x)=x^{2}+2, x \geq 0$$

19. $$f^{-1}(x)=x^{3}+3$$

21. $$f^{-1}(x)=\frac{x^{4}+5}{9}, x \geq 0$$

23. $$f^{-1}(x)=\frac{x^{5}-5}{-3}$$

##### Exercise $$\PageIndex{27}$$ Writing Exercises
1. Explain how the graph of the inverse of a function is related to the graph of the function.
2. Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.