Question 1

Which of the following statements best describes a vertical asymptote of a rational function?

Accepted Answer

It is a line that the graph approaches but never crosses, caused by the denominator being zero.

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

$y = 1$

Question 4

A rational function $f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$ has which of the following features?

Accepted Answer

A hole at $x = 2$ and a vertical asymptote at $x = 3$

Question 5

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

Accepted Answer

$y = \frac{5}{2}$

Question 6

Which of the following best describes a hole (removable discontinuity) in the graph of a rational function?

Accepted Answer

A point where both the numerator and denominator are zero after factoring.

Question 7

Given $f(x) = \frac{x^2-4}{x^2-5x+6}$, find all vertical asymptotes.

Accepted Answer

$x = 2$ and $x = 3$

Question 8

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

Accepted Answer

$y = 1$

Question 9

A rational function $f(x) = \frac{x^2-9}{x^2-4x+3}$ is graphed. Which $x$-values are vertical asymptotes?

Accepted Answer

$x = 3$ and $x = 1$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

$y = 3$

Question 12

A rational function $f(x) = \frac{x^2-4x+4}{x^2-4}$ has which of the following features?

Accepted Answer

A hole at $x = 2$ and a vertical asymptote at $x = -2$

Question 13

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

Accepted Answer

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

Question 14

Which of the following best describes the effect of vertical asymptotes on the domain of a rational function?

Accepted Answer

Vertical asymptotes restrict the domain of the function.

Question 15

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

Accepted Answer

$y = 2$

Question 16

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

Accepted Answer

$x = 3$ and $x = 1$

Question 17

A rational function $f(x) = \frac{x^2-4}{x^2-5x+6}$ has a hole at which $x$-value?

Accepted Answer

There is no hole in the graph.

Question 18

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

Accepted Answer

$y = 2$

Question 19

Which of the following is true about the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2+3}$?

Accepted Answer

$y = \frac{1}{2}$

Question 20

A rational function $f(x) = \frac{x^2-4x+4}{x^2-4}$ has a hole at $x = 2$. What is the $y$-value of the hole?

Accepted Answer

$y = 0$

Asymptotes of Rational Functions: Vertical, Horizontal, and Holes

Study Guide - Practice Questions