Question 1

Which of the following is the correct definition of a rational function?

Accepted Answer

A function that can be written as $$f(x) = \frac{p(x)}{q(x)}$$, where $$p(x)$$ and $$q(x)$$ are polynomials and $$q(x) 
eq 0.$$

Question 2

Given $$f(x) = \frac{x^2$ - 4}{x^2$ - 5x + 6}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq 2, x 
eq 3$$

Question 3

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

Accepted Answer

A line $$x = a$$ where the function approaches infinity as $$x$$ approaches $$a.$$

Question 4

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

Accepted Answer

$$x = 3$$

Question 5

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

Accepted Answer

$$y = 3$$

Question 6

Which transformation will shift the graph of $$y = f(x) up by 4 $$units?

Accepted Answer

$$y = f(x) + 4$$

Question 7

If $$f(x) = \frac{1}{x}$$, what is the effect of the transformation $$y = -2f(x)$$?

Accepted Answer

Reflect about the $$x-$$axis and stretch vertically by a factor of 2.

Question 8

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

Accepted Answer

$$(0, -2)$$

Question 9

A rational function has a hole at $$x = 2. $$Which of the following could be its equation?

Accepted Answer

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

Question 10

Given $$f(x) = \frac{x^2$ + 1}{x - 4}$$, what is the end behavior as $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 11

Which of the following rational functions has an oblique (slant) asymptote?

Accepted Answer

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

Question 12

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

Accepted Answer

$$x = -1, x = -0.5$$

Question 13

Which of the following best describes the effect of $$y = f(-x) on $$the graph of $$y = f(x)$$?

Accepted Answer

Reflects the graph about the $$y-$$axis.

Question 14

Given $$f(x) = \frac{x^2$ - 9}{x^2$ - 4}$$, which of the following statements is true?

Accepted Answer

There are vertical asymptotes at $$x = 2$$ and $$x = -2.$$

Question 15

If $$f(x) = \frac{1}{x-2}$$, what is the behavior of $$f(x) as$$ $$x$$ approaches 2 from the right?

Accepted Answer

$$f(x) 	o \infty$$

Question 16

Which of the following rational functions has no $$y-$$intercept?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 17

A rational function $$f(x)$$ has a vertical asymptote at $$x = -3$$ and a hole at $$x = 2. $$Which could be its equation?

Accepted Answer

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

Question 18

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

Accepted Answer

$$y = 2x + 3$$

Properties and Graphing of Rational Functions

Study Guide - Practice Questions