Question 1

Which of the following best describes the domain of a rational function $$f(x) = \frac{p(x)}{q(x)}$$?

Accepted Answer

All real numbers except where $$q(x) = 0$$

Question 2

Given the graph in the first image, what are the equations of the vertical and horizontal asymptotes?

Accepted Answer

Vertical: $$x = -6$$, $$x = 6$$; Horizontal: $$y = 0$$

Question 3

For the function $$f(x) = \frac{2x^2$ + 3}{x^2$ - 4}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 4

Which of the following functions is NOT a rational function?

Accepted Answer

$$f(x) = \sqrt{x} + \frac{1}{x}$$

Question 5

Given the function $$f(x) = \frac{5x}{x^2$ - 9}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 6

For the function $$f(x) = \frac{4x^2$ + 1}{2x^2$ - 3x + 5}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 7

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

Accepted Answer

The maximum population the farm can sustain as time goes to infinity

Question 8

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

Accepted Answer

All real numbers

Question 9

Given the second image, what is the equation of the vertical asymptote?

Accepted Answer

$$x = -4$$

Question 10

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

Accepted Answer

All real numbers except $$x = 1$$

Question 11

If the degree of the numerator of a rational function is less than the degree of the denominator, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 12

For the function $$f(x) = \frac{x^2$ + 2x + 1}{x^2$ - 1}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 13

A tree survival model is given by $$P(t) = \frac{300 - t}{t + 3}$$ for $$0 \leq t \leq 300. $$What is the horizontal asymptote as $$t 	o \infty$$?

Accepted Answer

$$y = -1$$

Question 14

For the function $$f(x) = \frac{x^3$ + 2x}{x^2$ - 1}$$, what can be said about the horizontal asymptote?

Accepted Answer

There is no horizontal asymptote

Question 15

If $$f(x) = \frac{2x^2$ + 3x + 1}{x^2$ - 4}$$, what is the domain of $$f$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 16

For the function $$f(x) = \frac{3x^2$ + 2x + 1}{x^2$ - 4x + 4}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 2$$

Question 17

If a rational function has a vertical asymptote at $$x = k$$, what does this mean for the function's graph?

Accepted Answer

The graph approaches infinity or negative infinity as $$x$$ approaches $$k$$

Question 18

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

Accepted Answer

$$y = 0$$

Question 19

If $$f(x) = \frac{4x^2$ + 1}{2x^2$ - 3x + 5}$$, what is the domain?

Accepted Answer

All real numbers

Rational Functions and Their Graphs: Asymptotes and Applications

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