Rational Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rational Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A rational function is one polynomial divided by another, behaving normally except where the denominator hits zero.

Common stuck point: The procedure for rational functions is the easy part; the trap is forgetting to exclude inputs where the denominator is zero. Asking "Is the function a polynomial divided by another polynomial containing the variable?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the function a polynomial divided by another polynomial containing the variable?

Worked Examples

Example 1

medium

Find the vertical and horizontal asymptotes of

f(x) = \frac{3x + 2}{x - 4}

Answer

\text{Vertical: } x = 4, \quad \text{Horizontal: } y = 3

First step

Vertical asymptote: set denominator

= 0

x - 4 = 0

, so

x = 4

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Example 2

hard

Find all asymptotes and holes of

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

Example 3

medium

Find all asymptotes of

f(x) = \dfrac{4x^2}{x^2 - 1}

Example 4

medium

Find the horizontal asymptote of

g(x) = \dfrac{3x^3 + x}{6x^3 - 5}

Example 5

medium

For

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

, find any holes and vertical asymptotes.

Example 6

hard

Find the slant asymptote of

f(x) = \dfrac{x^2 + 3x + 1}{x - 1}

Example 7

hard

Find the slant asymptote of

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

Example 8

hard

Combine

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

into a single rational expression.

Example 9

challenge

Find all values of

k

such that

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

has a hole at

x = 2

rather than a vertical asymptote.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Find the

x

-intercepts and vertical asymptotes of

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

Example 2

hard

Find all asymptotes (vertical, horizontal, and oblique) of

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

. Does the function have a hole or a vertical asymptote at

x = 1

Example 3

easy

Find the vertical asymptote of

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

Example 4

easy

Find the domain of

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

Example 5

easy

Find the

x

-intercept of

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

Example 6

easy

Evaluate

f(3)

for

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

Example 7

easy

Find the

y

-intercept of

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

Example 8

easy

What value must be excluded from

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

Example 9

easy

Find the horizontal asymptote of

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

Example 10

easy

Simplify

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

and state the restriction.

Example 11

medium

Find all vertical asymptotes of

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

Example 12

medium

Identify the hole of

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

Example 13

medium

Find the horizontal asymptote of

f(x)=\frac{2x^2+1}{x^2-5}

Example 14

medium

Find the horizontal asymptote of

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

Example 15

medium

Solve

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

Example 16

medium

Does

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

have a horizontal asymptote?

Example 17

medium

Find the slant asymptote of

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

Example 18

medium

Find the domain and any holes of

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

Example 19

challenge

Find the values of

x

where

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

has a hole versus an asymptote.

Example 20

challenge

Solve

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

Example 21

challenge

For what value of

k

does

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

have a hole at

x=3

Example 22

medium

Find the vertical asymptotes of

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

Example 23

easy

State the domain of

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

Example 24

easy

Find the vertical asymptote of

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

Example 25

easy

Find

f(2)

for

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

Example 26

easy

What value of

x

is excluded from

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

Example 27

medium

Simplify

\dfrac{x^2 - 4}{x^2 - 2x}

and state restrictions.

Example 28

medium

Identify the hole and vertical asymptote of

f(x) = \dfrac{(x-2)(x+5)}{(x-2)(x-3)}

Example 29

medium

Solve

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

Example 30

medium

Find the domain of

f(x) = \dfrac{x+1}{x^2 - 5x + 6}

Example 31

medium

Find all

x

-intercepts of

f(x) = \dfrac{x^2 - 4}{x + 5}

Example 32

medium

Find the value of

a

such that

f(x) = \dfrac{a x + 6}{x - 2}

has a horizontal asymptote

y = 5

Example 33

hard

Solve

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

Example 34

hard

Find all vertical asymptotes and holes of

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

Example 35

hard

Solve

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

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Related Concepts

Polynomial Functions Fractions Asymptote

Background Knowledge

These ideas may be useful before you work through the harder examples.

polynomial functionsfractions