$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 8.8E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

### Practice Makes Perfect

##### Exercise $$\PageIndex{17}$$ Evaluate a Radical Function

In the following exercises, evaluate each function.

1. $$f(x)=\sqrt{4 x-4}$$, find
1. $$f(5)$$
2. $$f(0)$$
2. $$f(x)=\sqrt{6 x-5}$$, find
1. $$f(5)$$
2. $$f(-1)$$
3. $$g(x)=\sqrt{6 x+1}$$, find
1. $$g(4)$$
2. $$g(8)$$
4. $$g(x)=\sqrt{3 x+1}$$, find
1. $$g(8)$$
2. $$g(5)$$
5. $$F(x)=\sqrt{3-2 x}$$, find
1. $$F(1)$$
2. $$F(-11)$$
6. $$F(x)=\sqrt{8-4 x}$$, find
1. $$F(1)$$
2. $$F(-2)$$
7. $$G(x)=\sqrt{5 x-1}$$, find
1. $$G(5)$$
2. $$G(2)$$
8. $$G(x)=\sqrt{4 x+1}$$, find
1. $$G(11)$$
2. $$G(2)$$
9. $$g(x)=\sqrt[3]{2 x-4}$$, find
1. $$g(6)$$
2. $$g(-2)$$
10. $$g(x)=\sqrt[3]{7 x-1}$$, find
1. $$g(4)$$
2. $$g(-1)$$
11. $$h(x)=\sqrt[3]{x^{2}-4}$$, find
1. $$h(-2)$$
2. $$h(6)$$
12. $$h(x)=\sqrt[3]{x^{2}+4}$$, find
1. $$h(-2)$$
2. $$h(6)$$
13. For the function $$f(x)=\sqrt[4]{2 x^{3}}$$, find
1. $$f(0)$$
2. $$f(2)$$
14. For the function $$f(x)=\sqrt[4]{3 x^{3}}$$, find
1. $$f(0)$$
2. $$f(3)$$
15. For the function $$g(x)=\sqrt[4]{4-4 x}$$, find
1. $$g(1)$$
2. $$g(-3)$$
16. For the function $$g(x)=\sqrt[4]{8-4 x}$$, find
1. $$g(-6)$$
2. $$g(2)$$

1.

1. $$f(5)=4$$
2. no value at $$x=0$$

3.

1. $$g(4)=5$$
2. $$g(8)=7$$

5.

1. $$F(1)=1$$
2. $$F(-11)=5$$

7.

1. $$G(5)=2 \sqrt{6}$$
2. $$G(2)=3$$

9.

1. $$g(6)=2$$
2. $$g(-2)=-2$$

11.

1. $$h(-2)=0$$
2. $$h(6)=2 \sqrt[3]{4}$$

13.

1. $$f(0)=0$$
2. $$f(2)=2$$

15.

1. $$g(1)=0$$
2. $$g(-3)=2$$
##### Exercise $$\PageIndex{18}$$ Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

1. $$f(x)=\sqrt{3 x-1}$$
2. $$f(x)=\sqrt{4 x-2}$$
3. $$g(x)=\sqrt{2-3 x}$$
4. $$g(x)=\sqrt{8-x}$$
5. $$h(x)=\sqrt{\frac{5}{x-2}}$$
6. $$h(x)=\sqrt{\frac{6}{x+3}}$$
7. $$f(x)=\sqrt{\frac{x+3}{x-2}}$$
8. $$f(x)=\sqrt{\frac{x-1}{x+4}}$$
9. $$g(x)=\sqrt[3]{8 x-1}$$
10. $$g(x)=\sqrt[3]{6 x+5}$$
11. $$f(x)=\sqrt[3]{4 x^{2}-16}$$
12. $$f(x)=\sqrt[3]{6 x^{2}-25}$$
13. $$F(x)=\sqrt[4]{8 x+3}$$
14. $$F(x)=\sqrt[4]{10-7 x}$$
15. $$G(x)=\sqrt[5]{2 x-1}$$
16. $$G(x)=\sqrt[5]{6 x-3}$$

1. $$\left[\frac{1}{3}, \infty\right)$$

3. $$\left(-\infty, \frac{2}{3}\right]$$

5. $$(2, \infty)$$

7. $$(-\infty,-3] \cup(2, \infty)$$

9. $$(-\infty, \infty)$$

11. $$(-\infty, \infty)$$

13. $$\left[-\frac{3}{8}, \infty\right)$$

15. $$(-\infty, \infty)$$

##### Exercise $$\PageIndex{19}$$ graph radical functions

In the following exercises,

1. find the domain of the function
2. graph the function
3. use the graph to determine the range
1. $$f(x)=\sqrt{x+1}$$
2. $$f(x)=\sqrt{x-1}$$
3. $$g(x)=\sqrt{x+4}$$
4. $$g(x)=\sqrt{x-4}$$
5. $$f(x)=\sqrt{x}+2$$
6. $$f(x)=\sqrt{x}-2$$
7. $$g(x)=2 \sqrt{x}$$
8. $$g(x)=3 \sqrt{x}$$
9. $$f(x)=\sqrt{3-x}$$
10. $$f(x)=\sqrt{4-x}$$
11. $$g(x)=-\sqrt{x}$$
12. $$g(x)=-\sqrt{x}+1$$
13. $$f(x)=\sqrt[3]{x+1}$$
14. $$f(x)=\sqrt[3]{x-1}$$
15. $$g(x)=\sqrt[3]{x+2}$$
16. $$g(x)=\sqrt[3]{x-2}$$
17. $$f(x)=\sqrt[3]{x}+3$$
18. $$f(x)=\sqrt[3]{x}-3$$
19. $$g(x)=\sqrt[3]{x}$$
20. $$g(x)=-\sqrt[3]{x}$$
21. $$f(x)=2 \sqrt[3]{x}$$
22. $$f(x)=-2 \sqrt[3]{x}$$

1.

1. domain: $$[-1, \infty)$$

2. Figure 8.7.8
3. $$[0, \infty)$$

3.

1. domain: $$[-4, \infty)$$

2. Figure 8.7.9
3. $$[0, \infty)$$

5.

1. domain: $$[0, \infty)$$

2. Figure 8.7.10
3. $$[2, \infty)$$

7.

1. domain: $$[0, \infty)$$

2. Figure 8.7.11
3. $$[0, \infty)$$

9.

1. domain: $$(-\infty, 3]$$

2. Figure 8.7.12
3. $$[0, \infty)$$

11.

1. domain: $$[0, \infty)$$

2. Figure 8.7.13
3. $$(-\infty, 0]$$

13.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.7.14
3. $$(-\infty, \infty)$$

15.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.7.15
3. $$(-\infty, \infty)$$

17.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.7.16
3. $$(-\infty, \infty)$$

19.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.7.17
3. $$(-\infty, \infty)$$

21.

1. domain: $$(-\infty, \infty)$$

2. Figure 8.7.18
3. $$(-\infty, \infty)$$
##### Exercise $$\PageIndex{20}$$ writing exercises
1. Explain how to find the domain of a fourth root function.
2. Explain how to find the domain of a fifth root function.
3. Explain why $$y=\sqrt[3]{x}$$ is a function.
4. Explain why the process of finding the domain of a radical function with an even index is different from the process when the index is odd.