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

스터디 가이드 - 연습 문제