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

$$\{x\ |\ x \neq 5,\ x \neq -5\}$$

Q: Given $$f(x) = \frac{x^2$-4x+1}{3x-2}$$, what is the domain of $$f(x)$$?

$$\{x\ |\ x \neq \frac{2}{3}\}$$

Q: Given $$f(x) = \frac{3x+12}{x^2$-25}$$, write the function in lowest terms and state its domain.

$$f(x) = \frac{3(x+4)}{(x+5)(x-5)}$$, domain: $$\{x\ |\ x \neq 5,\ x \neq -5\}$$

Question 1

Which of the following best describes a rational function?

Accepted Answer

A function that can be written as the ratio of two polynomials.

Question 2

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

Accepted Answer

$$\{x\ |\ x 
eq 5,\ x 
eq -5\}$$

Question 3

Given $$f(x) = \frac{x^2$-4x+1}{3x-2}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$\{x\ |\ x 
eq \frac{2}{3}\}$$

Question 4

Find the vertical asymptote(s) of $$f(x) = \frac{x+2}{(x+2)(x-3)}.$$

Accepted Answer

$$x = 3$$

Question 5

Which of the following statements about horizontal asymptotes of rational functions is TRUE?

Accepted Answer

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$y=0.$$

Question 6

Determine the horizontal asymptote of $$f(x) = \frac{2x^2$}{3x^2$ + x - 1}.$$

Accepted Answer

$$y = \frac{2}{3}$$

Question 7

Find all vertical asymptotes and holes for $$f(x) = \frac{x^2$-1}{x-1}.$$

Accepted Answer

Hole at $$x=1$$; no vertical asymptote.

Question 8

Given $$f(x) = \frac{3x+12}{x^2$-25}$$, write the function in lowest terms and state its domain.

Accepted Answer

$$f(x) = \frac{3(x+4)}{(x+5)(x-5)}$$, domain: $$\{x\ |\ x 
eq 5,\ x 
eq -5\}$$

Question 9

A rational function has a removable discontinuity (hole) at $$x=2. $$Which of the following could be its formula?

Accepted Answer

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

Question 10

Which of the following best describes the end behavior of $$f(x) = \frac{1}{x^2$} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 0$$

Question 11

Suppose a rational function models the concentration $$C(t) of a $$drug in the bloodstream over time $$t (in $$hours): $$C(t) = \frac{10}{t+2}. $$What is the horizontal asymptote and what does it represent in this context?

Accepted Answer

Horizontal asymptote $$y=0$$; as time increases, the drug concentration approaches zero.

Question 12

Given $$f(x) = \frac{2x-3}{x-1}$$, find the $$x-$$intercept(s) of the function.

Accepted Answer

$$x = \frac{3}{2}$$

Question 13

A rational function $$f(x)$$ has a vertical asymptote at $$x=4$$ and a hole at $$x=-2. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = \frac{(x+2)(x-1)}{(x-4)(x+2)}$$

Question 14

Which transformation is applied to the parent function $$f(x) = \frac{1}{x} to $$obtain $$g(x) = \frac{1}{x-3} + 1$$?

Accepted Answer

Shift right by 3 units and up by 1 unit.

Question 15

Find the $$y-$$intercept of $$f(x) = \frac{x^2$-4}{x-2}.$$

Accepted Answer

$$y = -2$$

Question 16

A rational function $$f(x)$$ has a horizontal asymptote at $$y=0$$ and a vertical asymptote at $$x=2. $$Which of the following could be $$f(x)$$?

Accepted Answer

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

Question 17

Given $$f(x) = \frac{x^2$-9}{x^2$-4x+3}$$, find all vertical asymptotes.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 18

A rational function $$f(x)$$ has a numerator and denominator of equal degree. What determines the horizontal asymptote?

Accepted Answer

The ratio of the leading coefficients.

Question 19

Suppose $$f(x) = \frac{2x^2$+3}{x^2$-6}. $$What is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Rational Functions: Concepts, Properties, and Graphing

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