Question 1

Which of the following is the correct domain for the function $$f(x) = \dfrac{x^2$ + 8}{x - 2}$$?

Accepted Answer

$$x 
eq 2$$

Question 2

Given $$f(x) = \dfrac{x^2$ + 6x + 8}{x^2$ - 2x - 8}$$, which values must be excluded from the domain?

Accepted Answer

$$x = 4$$ and $$x = -2$$

Question 3

As $$x$$ approaches $$2$$ from the right in $$f(x) = \dfrac{x^2$ + 8}{x - 2}$$, what happens to $$f(x)$$?

Accepted Answer

$$f(x) 	o +\infty$$

Question 4

Which of the following best describes the behavior of $$\dfrac{1}{x} as$$ $$x$$ approaches $$0$$ from the left?

Accepted Answer

$$\dfrac{1}{x} 	o -\infty$$

Question 5

For $$f(x) = \dfrac{x^2$ - 4}{x^2$ - 4}$$, what is true about the graph?

Accepted Answer

There is a hole at $$x = 2$$ and $$x = -2$$; no vertical asymptotes.

Question 6

Find the vertical asymptotes and holes for $$f(x) = \dfrac{x^2$ - 4}{x^2$ - 2x - 8}.$$

Accepted Answer

Vertical asymptotes at $$x = 4$$ and $$x = -2$$; no holes.

Question 7

Given $$f(x) = \dfrac{x^2$ - 25}{x + 5}$$, what is the vertical asymptote?

Accepted Answer

There is no vertical asymptote; there is a hole at $$x = -5.$$

Question 8

Which of the following describes the horizontal asymptote of $$f(x) = \dfrac{3x^2$ + 15}{x + 1}$$?

Accepted Answer

There is no horizontal asymptote.

Question 9

For $$f(x) = \dfrac{2x^2$ + 15}{3x^2$ + 1}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \dfrac{2}{3}$$

Question 10

A rational function has a numerator of degree 3 and a denominator of degree 2. What type of asymptote does it have?

Accepted Answer

A slant (oblique) asymptote.

Question 11

Find the equation of the slant asymptote for $$f(x) = \dfrac{x^2$ + 1}{x - 1}.$$

Accepted Answer

$$y = x + 1$$

Question 12

Which of the following steps is NOT necessary when graphing a rational function?

Accepted Answer

Find the maximum value of the function.

Question 13

Given $$f(x) = \dfrac{x^2$ - 4x + 4}{x^2$ - 6x + 9}$$, what is the behavior of the graph?

Accepted Answer

There is a hole at $$x = 3$$; otherwise, the graph is $$y = 1.$$

Question 14

A chemical reaction's rate is modeled by $$R(x) = \dfrac{5x}{x - 2}. $$What happens to the rate as the concentration $$x$$ approaches $$2$$ from below?

Accepted Answer

$$R(x) 	o -\infty$$

Question 15

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

Accepted Answer

$$y = 1$$

Question 16

Which of the following rational functions has a hole at $$x = 2$$?

Accepted Answer

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

Question 17

A rational function models the spread of a disease: $$S(t) = \dfrac{100}{t - 5}. $$What is the significance of $$t = 5$$?

Accepted Answer

There is a vertical asymptote at $$t = 5$$; the model predicts an infinite rate at this time.

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

$$f(x) = \dfrac{4x}{x^2$ - 1}$$

Question 20

Suppose $$f(x) = \dfrac{x^2$ - 1}{x - 1}. $$What is the simplified form and what feature does the graph have at $$x = 1$$?

Accepted Answer

Simplifies to $$x + 1$$ with a hole at $$x = 1.$$

Rational Functions and Their Graphs: Study Guide

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