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 polynomial functions and $$q(x) 
eq 0.$$

Question 2

What is the domain of $$f(x) = \frac{x^2$ - 25}{x - 5}$$?

Accepted Answer

$$(-\infty, 5) \cup (5, \infty)$$

Question 3

Given $$g(x) = \frac{x}{x^2$ - 25}$$, which values must be excluded from the domain?

Accepted Answer

$$x = 5$$ and $$x = -5$$

Question 4

Which of the following rational functions has a domain of all real numbers?

Accepted Answer

$$f(x) = \frac{x+5}{x^2$ + 25}$$

Question 5

What is the end behavior of $$f(x) = \frac{1}{x} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 0$$

Question 6

Which statement best describes the vertical asymptote(s) of $$f(x) = \frac{x}{x^2$ - 1}$$?

Accepted Answer

There are vertical asymptotes at $$x = 1$$ and $$x = -1.$$

Question 7

After simplifying $$g(x) = \frac{x-1}{x^2$-1}$$, what is the vertical asymptote?

Accepted Answer

$$x = -1$$

Question 8

Which of the following rational functions has no vertical asymptotes?

Accepted Answer

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

Question 9

What is the horizontal asymptote of $$f(x) = \frac{9x^2$}{3x^2$ + 1}$$?

Accepted Answer

$$y = 3$$

Question 10

For $$g(x) = \frac{9x}{3x^2$ + 1}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 11

For $$h(x) = \frac{9x^3$}{3x^2$ + 1}$$, what type of asymptote exists?

Accepted Answer

Slant asymptote at $$y = 3x$$

Question 12

What is the equation of the slant asymptote for $$f(x) = \frac{x^2$ + 1}{x - 1}$$?

Accepted Answer

$$y = x + 1$$

Question 13

Which transformation describes $$g(x) = \frac{1}{x+2} - 1$$ compared to $$f(x) = \frac{1}{x}$$?

Accepted Answer

Shifted 2 units left and 1 unit down

Question 14

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

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 15

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

Accepted Answer

$$y = 3$$

Question 16

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

Accepted Answer

Even function, symmetric about the y-axis

Question 17

What is the $$y-$$intercept of $$f(x) = \frac{3x^2$}{x^2$ - 4}$$?

Accepted Answer

$$y = 0$$

Question 18

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

Accepted Answer

It has a vertical asymptote at $$x = 0$$ and is always positive for all $$x 
eq 0.$$

Question 19

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a hole at $$x = a if$$:

Accepted Answer

$$p(a) = 0$$ and $$q(a) = 0$$ after simplification$

Rational Functions: Domains, Asymptotes, and Graphing

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