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Exercises 8.5 Practice Problems

1.

Write each of the following in interval notation (Note: you may find it helpful to draw a number line):
  1. \(\displaystyle x \ge -5\)
  2. \(x \le -3\) or \(x \gt 6\)
  3. \(\displaystyle x \neq 88\)
  4. \(x \ge -2\) and \(x \lt 6\)
  5. Everything except \(\pm 6\)
  6. \(\displaystyle x \lt 18\)
  7. Everything except \(0\)
  8. \(x \lt -5\) or \(x \gt -5\)
Answer.
  1. \(\displaystyle [-5, \infty)\)
  2. \(\displaystyle (-\infty,-3]\cup(6,\infty)\)
  3. \(\displaystyle (-\infty,88)\cup(88,\infty)\)
  4. \(\displaystyle [-2, 6)\)
  5. \(\displaystyle (-\infty,-6)\cup(-6,6)\cup(6,\infty)\)
  6. \(\displaystyle (-\infty,18)\)
  7. \(\displaystyle (-\infty,0)\cup(0,\infty)\)
  8. \(\displaystyle (-\infty, -5)\cup(-5, \infty)\)

2.

Find the domain of the function in the graph below:
Answer.
\([1,3)\)

3.

Find the domain of the function in the graph below:
Answer.
\([1,4)\cup(4,6]\)

4.

Find the domain of the function in the graph below:
Answer.
\([1,9)\)

5.

Find the domain of the function in the graph below:
Answer.
\([-3,1)\cup[2,4]\)

6.

Find the domain of the function in the graph below:
Answer.
\([-1,6]\)

7.

Find the domain of the function in the graph below:
Answer.
\([-2,0]\cup(1,3)\cup[5,7]\)

8.

Find the domain of the function in the graph below:
Answer.
\([-2,0]\cup(2,5)\)

9.

Find the domain of the function in the graph below:
Answer.
\([-1,2)\cup(2,5]\)

10.

Find the range of the function in the graph below:
Answer.
\([0,4)\)

11.

Find the range of the function in the graph below:
Answer.
\([0,9)\)

12.

Find the range of the function in the graph below:
Answer.
\([-1,7)\)

13.

Find the range of the function in the graph below:
Answer.
\([0,9]\)

14.

Find the range of the function in the graph below:
Answer.
\([0,1]\cup(2,7]\)

15.

Find the range of the function in the graph below:
Answer.
\([-2,0]\cup(1,3)\cup[5,7]\)

16.

Find the range of the function in the graph below:
Answer.
\([4,14)\)

17.

Find the range of the function in the graph below:
Answer.
\([-4,2)\cup(2,8]\)

18.

Find the domain of \(f(x)=5x^2-\frac{7x}{2}+\sqrt{29}\text{.}\)
Answer.
\((-\infty, \infty)\)

19.

Find the domain of \(f(x)=\frac{3x^2}{2x-1}\text{.}\)
Answer.
\((-\infty, \frac{1}{2})\cup(\frac{1}{2},\infty)\)

20.

Find the domain of \(f(x)=\sqrt{4x-3}\text{.}\)
Answer.
\([\frac{3}{4}, \infty)\)

21.

Find the domain of \(f(x)=\frac{4x^3}{\sqrt{3x}}\text{.}\)
Answer.
\((0, \infty)\)

22.

Find the domain of \(g(x)=7x^2-\frac{7x}{\sqrt{7}}+\sqrt{31}\text{.}\)
Answer.
\((-\infty, \infty)\)

23.

Find the domain of \(h(x)=15x^6-\frac{19x}{x^2-1}\text{.}\)
Answer.
\((-\infty,-1)\cup(-1,1)\cup(1,\infty)\)

24.

Find the domain of \(f(x)=\frac{\sqrt{2x+3}}{4}\text{.}\)
Answer.
\([-\frac{3}{2}, \infty)\)

25.

Find the domain of \(g(x)=\frac{1}{\sqrt{6x+4}}\text{.}\)
Answer.
\((-\frac{2}{3}, \infty)\)

26.

Find the domain of \(f(x)=13x^{12}-\frac{\pi x}{11}+\sqrt{10}\text{.}\)
Answer.
\((-\infty, \infty)\)

27.

Find the domain of \(f(x)=\frac{x-3}{12x+18}\text{.}\)
Answer.
\((-\infty, -\frac{3}{2})\cup(-\frac{3}{2}, \infty)\)
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