Q: For the function $$f(x) = \frac{1}{3x - 5}$$, what is the domain in set notation?

$$\{ x \in \mathbb{R} \mid x \neq \frac{5}{3} \}$$

Q: For the function $$y = \frac{\sqrt{4 - x}}{\sqrt{5 - x}}$$, what is the domain?

$$\{ x \in \mathbb{R} \mid x \u003c 4 \text{ and } x \u003c 5 \}$$

Question 1

Given the piecewise function $$g(x)$$ defined as:

$$\begin{cases}
-3x^2$ + 8, & x \u003c -2 \
\frac{x^2$ - 4}{x + 2}, & -2 \leq x \u003c 1 \
5 - 2x, & x \geq 1
\end{cases}$$

What is $$g(-3)$$?

Accepted Answer

$$-3(-3)^2 + 8 = -27 + 8 = -19$

Question 2

For the function $$f(x) = \frac{1}{3x - 5}$$, what is the domain in set notation?

Accepted Answer

$$\{ x \in \mathbb{R} \mid x 
eq \frac{5}{3} \}$$

Question 3

Given $$f(x) = \sqrt{3 - 7x}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$\{ x \in \mathbb{R} \mid x \leq \frac{3}{7} \}$$

Question 4

For the function $$f(x) = 6x^3$ - 4x - 1$$, what is the domain?

Accepted Answer

$$\{ x \in \mathbb{R} \}$$

Question 5

Find the zeros of $$y = 5 - \sqrt{8x}.$$

Accepted Answer

$$x = \frac{25}{8}$$

Question 6

Which of the following is NOT a function?

Accepted Answer

$$3x - 2y = 8xy$$

Question 7

Given $$f(x) = x^2$ + 22x - 8x^3$, what is the y-intercept?

Accepted Answer

$$f(0) = 0^2$ + 22 \cdot 0 - 8 \cdot 0^3$ = 0$$

Question 8

For the function $$f(x) = \frac{1}{\sqrt{x + 3}}$$, what is the domain?

Accepted Answer

$$\{ x \in \mathbb{R} \mid x \u003e -3 \}$$

Question 9

Given $$y = |4x - 1|$$, what is the x-intercept?

Accepted Answer

$$x = \frac{1}{4}$$

Question 10

For the function $$y = \frac{\sqrt{4 - x}}{\sqrt{5 - x}}$$, what is the domain?

Accepted Answer

$$\{ x \in \mathbb{R} \mid x \u003c 4 	ext{ and } x \u003c 5 \}$$

Question 11

Given the piecewise function $$f(x) = \begin{cases} -\frac{1}{2}x - 1, & x \u003c -6 \ -6, & -6 \leq x \u003c 1 \ x, & x \geq 1 \end{cases}$$, what is $$f(-7)$$?

Accepted Answer

$$-\frac{1}{2}(-7) - 1 = 3.5 - 1 = 2.5$$

Question 12

For the function $$f(x) = \frac{x - 5}{x^2$ + 5}$$, what is the domain?

Accepted Answer

$$\{ x \in \mathbb{R} \}$$

Question 13

Given $$g(x) = 5 - 2x$$ for $$x \geq 1$$, what is $$g(7)$$?

Accepted Answer

$$5 - 2 \cdot 7 = 5 - 14 = -9$$

Question 14

For the function $$f(x) = \sqrt[3]{x^2$ - 5x - 2}$$, what is the domain?

Accepted Answer

$$\{ x \in \mathbb{R} \}$$

Question 15

Given $$f(x) = 3x^2$ - 5x - 7$$, what is $$f(-3)$$?

Accepted Answer

$$3(-3)^2 - 5(-3) - 7 = 27 + 15 - 7 = 35$

Question 16

For the function $$f(x) = \frac{x + 2}{3x^2$ - 5x - 2}$$, what values of $$x$$ are excluded from the domain?

Accepted Answer

$$x = 2$$ and $$x = -\frac{1}{3}$$

Question 17

Given the function $$f(x)$$ graphed in image 1, what is the range in interval notation?

Accepted Answer

$$(-\infty, 4]$$

Question 18

For the function $$f(x) = x^2$ + 22x - 8x^3$, what are the zeros?

Accepted Answer

$$x = 0, x = \frac{22 \pm \sqrt{22^2$ - 4 \cdot 1 \cdot (-8)}}{2 \cdot (-8)}$$

Study Guidance for College Calculus: Functions, Domain, Range, and Related Concepts

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