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All courses Algebra II · F-IF.5 29 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

Apply the denominator rule

\[x-4\ne~0\]

A rational function is defined only when its denominator is nonzero.

02

Solve the restriction

\[x-4\ne~0~->~x\ne~4\]

The only denominator zero is x=4, so exclude that input and retain every other real number.

03

State the domain

\[x~\ne~4\]

The domain is all real x except 4, which is choice A.

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Another way

  1. Write the domain as (-infinity,4) union (4,infinity).

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Common mistake

Do not set the numerator equal to zero for a domain question; restrictions come from denominator zeros.

Solution walkthrough video