$$f(x) \sim 3x as$$ $$x \to \infty$$

Question 1

Given the graph below, which of the following is the equation of the horizontal asymptote?

(Refer to the first image provided.)

Accepted Answer

$$y = 2$$

Question 2

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

Accepted Answer

All real numbers $$x$$

Question 3

Which of the following rational functions has a vertical asymptote at $$x = 3$$?

Accepted Answer

$$f(x) = \frac{4x}{x-3}$$

Question 4

A rational function $$f(x) = \frac{2x^3$ - 4x}{x^2$ + x - 3} is $$shown in the third image. What is the equation of its oblique (slant) asymptote?

Accepted Answer

$$y = 2x - 2$$

Question 5

Which of the following statements is TRUE about the graph of a rational function?

Accepted Answer

The graph can cross a horizontal asymptote but cannot cross a vertical asymptote.

Question 6

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

Accepted Answer

$$(-\infty, 1) \cup (1, \infty)$$

Question 7

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

Accepted Answer

$$y = 1$$

Question 8

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

Accepted Answer

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

Question 9

If $$f(x) = \frac{3x^n + 2x - 1}{x^3$ - 1}$$, for which value of $$n$$ does $$f(x)$$ have a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$n \u003c 3$$

Question 10

Given $$f(x) = \frac{x^2$ - 9}{x^2$ + 4x - 21}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 3$$ and $$x = -7$$

Question 11

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

Accepted Answer

$$x = 0$$

Question 12

Which of the following best describes the end behavior of $$f(x) = \frac{3x^4$ - x^2$}{x^3$ - x^2$ + 1}$$?

Accepted Answer

$$f(x) \sim 3x as$$ $$x 	o \infty$$

Question 13

Given $$f(x) = \frac{x^2$ + ax}{x^2$ - 1}$$, for what value of $$a$$ does $$f(x)$$ have a hole at $$x = a$$?

Accepted Answer

$$a = 1 or$$ $$a = -1$$

Question 14

Which of the following rational functions has a horizontal asymptote at $$y = 3$$?

Accepted Answer

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

Question 15

If $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is the simplified form of $$f(x)$$ for $$x 
eq 1$$?

Accepted Answer

$$x + 1$$

Question 16

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

Accepted Answer

$$y = -4$$

Question 17

Which of the following best describes the process to find the vertical asymptotes of a rational function?

Accepted Answer

Set the denominator equal to zero and solve for $$x.$$

Question 18

Given $$f(x) = \frac{x^2$ - 9}{x^2$ + 4x - 21}$$, what is the $$y-$$intercept?

Accepted Answer

$$y = -\frac{9}{21}$$

Question 19

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

Accepted Answer

$$y = 0$$

Study Notes: Properties and Graphs of Rational Functions

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