Practice Rational Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A rational function is a ratio of two polynomials: $f(x) = P(x)/Q(x)$ where $P$ and $Q$ are polynomials and $Q(x) \neq 0$ .

Rational functions are the "fractions" of the function world — they behave like polynomials except near the zeros of the denominator, where they blow up or have holes.

Showing a random 20 of 50 problems.

Example 1

challenge

Solve

\frac{1}{x-1}+\frac{1}{x+1}=\frac{2}{x^2-1}

Example 2

hard

Find all vertical asymptotes and holes of

f(x) = \dfrac{x^2 - x - 6}{x^2 - 4}

Example 3

easy

State the domain of

f(x) = \dfrac{x+2}{x-7}

Example 4

easy

Find the horizontal asymptote of

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

Example 5

medium

Solve

\dfrac{x+3}{x-1} = 2

Example 6

easy

Find the vertical asymptote of

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

Example 7

medium

Solve

\frac{x+1}{x-2}=3

Example 8

hard

Find all asymptotes and holes of

f(x) = \frac{x^2 - 9}{x^2 - x - 6}

Example 9

medium

Find the slant asymptote of

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

Example 10

easy

What value must be excluded from

f(x)=\frac{3}{2x-6}

Example 11

medium

Find the

x

-intercepts and vertical asymptotes of

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

Example 12

challenge

For what value of

k

does

f(x)=\frac{x^2-k}{x-3}

have a hole at

x=3

Example 13

easy

Find the

x

-intercept of

f(x)=\frac{x-5}{x+2}

Example 14

easy

What value of

x

is excluded from

f(x) = \dfrac{5}{x^2 - 9}

Example 15

medium

For

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

, find any holes and vertical asymptotes.

Example 16

medium

Identify the hole of

f(x)=\frac{x^2-4}{x-2}

Example 17

easy

Find the

x

-intercept of

f(x) = \dfrac{2x - 8}{x + 3}

Example 18

easy

Find the

y

-intercept of

f(x)=\frac{x+6}{x-3}

Example 19

hard

For

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

, find the slant asymptote.

Example 20

medium

Does

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

have a horizontal asymptote?

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Related Concepts

Polynomial Functions Fractions Asymptote