Q: Which of the following is the correct domain for the rational function $$f(x) = \frac{2x+1}{x^2$-9}$$?

$$\{x \mid x \neq 3, x \neq -3\}$$

Question 1

Which of the following is the correct domain for the rational function $$f(x) = \frac{2x+1}{x^2$-9}$$?

Accepted Answer

$$\{x \mid x 
eq 3, x 
eq -3\}$$

Question 2

Given the graph below, which equations represent the vertical and horizontal asymptotes?

(Refer to the first image provided.)

Accepted Answer

Vertical: $$x = -6$$, $$x = 6$$; Horizontal: $$y = 2$$

Question 3

Which of the following functions has a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{3}{x^2$+1}$$

Question 4

For the function $$f(x) = \frac{5x-4}{x+2}$$, what is the equation of the vertical asymptote?

Accepted Answer

$$x = -2$$

Question 5

A population of rabbits is modeled by $$P(t) = \frac{40t+20}{t+5}$$, where $$t is $$the number of months after 4 rabbits are introduced. What does the horizontal asymptote represent in this context?

Accepted Answer

The maximum population the rabbits can reach as time increases.

Question 6

Given $$f(x) = \frac{x^2$-4}{x-2}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$\{x \mid x 
eq 2\}$$

Question 7

Which of the following rational functions has no horizontal asymptote?

Accepted Answer

$$f(x) = \frac{x^3$+1}{x^2$-1}$$

Question 8

For the function $$f(x) = \frac{2x^2$+3}{x^2$-1}$$, what is the equation of the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 9

Given the graph below, which function could represent it?

(Refer to the second image provided.)

Accepted Answer

$$f(x) = \frac{-4}{x}$$

Question 10

Suppose the percentage of a certain type of tree surviving to age $$t is $$modeled by $$P(t) = \frac{300-t}{t+3}$$ for $$0 \leq t \leq 300. $$What is the horizontal asymptote of this function?

Accepted Answer

$$y = -1$$

Question 11

If $$f(x) = \frac{x^2$-9}{x^2$-4}$$, what are the equations of the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 12

Which of the following statements is true about the rational function $$f(x) = \frac{3x+2}{x-1}$$?

Accepted Answer

It has a vertical asymptote at $$x = 1$$ and a horizontal asymptote at $$y = 3$$

Question 13

A rational function $$f(x)$$ has a vertical asymptote at $$x = 4$$ and a horizontal asymptote at $$y = 0. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = \frac{2}{x-4}$$

Question 14

If $$f(x) = \frac{4x}{x^2$+1}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 15

Which of the following best describes the end behavior of $$f(x) = \frac{5x^2$+1}{2x^2$+3} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \frac{5}{2}$$

Rational Functions and Their Graphs: Asymptotes and Applications

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