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All courses Math III · F-IF.5 22 of 55
Relate a function's domain to its equation, graph, and context, especially when model choice matters.

Find the domain of a rational function by excluding denominator zeros

Problem
What is the domain of f(x)=1/(x-4)?
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Hint

Set the denominator \(x-4\) equal to \(0\) to find the value that must be excluded.

A rational expression is undefined when its denominator is \(0\), so denominator values that make \(0\) create restrictions.

Solution walkthrough

01

Identify the denominator

\[x-4\]

The denominator is the part that controls when the rational expression is defined.

02

Set it equal to zero

\[x-4~=~0\]

To find restricted values, solve for when the denominator would become zero.

03

Solve for the restricted value

\[x~=~4\]

This is the value that makes the denominator zero.

04

Write the domain

\[x~\ne~4\]

All real x-values are allowed except \(4\), so the domain is all real \(x\) except \(4\).

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Another way

  1. You can check directly: substituting \(x=4\) gives \(1/(4-4)=1/0\), which is undefined.

!

Common mistake

A common mistake is to say the domain is \(x=4\), but \(4\) is the excluded value, not the allowed one.