Question 1

Find all asymptotes (horizontal and vertical) of $$f(x) = \frac{x-2}{x^2$-4}.$$

Accepted Answer

$$x = 2$$, $$x = -2$$ and $$y = 0$$

Question 2

Find the horizontal asymptote (HA) and vertical asymptote (VA) of $$f(x) = \frac{x^2$-4}{x(x+2)}.$$

Accepted Answer

HA: $$y = 1$$ \quad VA: $$x = 0, x = -2$$

Question 3

Find the oblique asymptote of $$f(x) = \frac{x^2$ + 1}{x - 1}.$$

Accepted Answer

$$y = x - 1$$

Question 4

Which of the following best describes the domain of $$f(x) = \frac{2x+8}{5x^2$+9x-2}$$?

Accepted Answer

All real numbers except $$x = -2$$ and $$x = \frac{1}{5}$$

Question 5

Given $$f(x) = \frac{4x + 3x^3$ - 7}{-3x + 4x^3$ + 2}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \frac{3}{4}$$

Question 6

A rational function has a numerator of degree 2 and a denominator of degree 1. What type of asymptote does it have?

Accepted Answer

An oblique asymptote

Question 7

For $$f(x) = \frac{x^2$ - 4}{(x-2)(14x-2)}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = \frac{1}{7}$$

Question 8

If $$f(x) = \frac{p(x)}{q(x)}$$ and $$\deg(p) \u003c \deg(q)$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 9

Which value(s) of $$x$$ create a hole in the graph of $$f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$$?

Accepted Answer

$$x = 2$$

Question 10

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

Accepted Answer

$$y = 3$$

Question 11

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has $$\deg(p) = \deg(q) + 1. $$What is the asymptote?

Accepted Answer

An oblique asymptote given by the quotient of $$p(x)$$ divided by $$q(x)$$

Question 12

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

Accepted Answer

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

Study Guide: Asymptotes and Domains of Rational Functions

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