Question 1

Which of the following best describes a rational function?

Accepted Answer

A function that is the ratio of two polynomials.

Question 2

Given the parent rational function $$f(x) = \frac{1}{x}$$, what are its vertical and horizontal asymptotes?

Accepted Answer

Vertical: $$x = 0$$, Horizontal: $$y = 0$$

Question 3

For the function $$f(x) = \frac{1}{x-2}$$, which transformation has been applied to the parent function $$f(x) = \frac{1}{x}$$?

Accepted Answer

A horizontal shift right by 2 units.

Question 4

What is the domain of the function $$f(x) = \frac{1}{x^2$-4}$$?

Accepted Answer

$$x 
eq 2$$ and $$x 
eq -2$$

Question 5

Which of the following functions has a hole in its graph?

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

$$y = 3$$

Question 8

Which scenario will result in an oblique (slant) asymptote for a rational function $$f(x) = \frac{p(x)}{q(x)}$$?

Accepted Answer

Degree of numerator is exactly one more than degree of denominator.

Question 9

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

Accepted Answer

Question 10

A rational function models the concentration of a drug in the bloodstream over time: $$C(t) = \frac{5t}{t^2$+1}. $$What happens to the concentration as $$t$$ becomes very large?

Accepted Answer

It approaches 0.

Question 11

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

Accepted Answer

$$x = -1$$

Question 12

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

Accepted Answer

$$y = 2$$

Question 13

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

Accepted Answer

Vertical shift up by 1 unit.

Question 14

If $$f(x) = \frac{x^2$-4}{x-2}$$, what is the simplified form and what type of discontinuity does it have?

Accepted Answer

Simplifies to $$x+2$$ with a hole at $$x=2.$$

Question 15

A rational function models the speed of a chemical reaction: $$v(t) = \frac{4t}{t^2$+4}. $$What is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 16

For $$f(x) = \frac{2x^2$+3x+2}{x+1}$$, use long division to find the oblique asymptote.

Accepted Answer

$$y = 2x + 1$$

Question 17

Which of the following best describes the behavior of $$f(x) = \frac{1}{x} as$$ $$x$$ approaches 0 from the right?

Accepted Answer

It approaches $$+\infty.$$

Question 18

Given $$f(x) = \frac{x^2$+1}{x^2$-1}$$, what are the vertical asymptotes?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 19

A rational function $$f(x) = \frac{ax^2$+bx+c}{dx^2$+ex+f}$$ models the population growth of a species. If $$a \u003c d$$, what is the long-term population trend?

Accepted Answer

Population approaches zero.

Rational Functions and Asymptotes: Precalculus Study Notes (Section 3.3)

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