How to Find the Domain of a Function

Precalculus · Functions and domains

How to Find the Domain of a Function

Problem

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

Answer

x \neq 4

The domain is the set of inputs that give a real output. Two things are forbidden: a denominator of zero and a square root of a negative number.

Step-by-step solution

Find the value that makes the denominator zero.
$x - 4 = 0$
Why: Division by zero is undefined, so the denominator cannot be zero.
Solve the small equation.
$x = 4$
Why: This input gives 1/0, which is not a real number.
Write the domain as every other real number.
$x \neq 4$
Why: All other inputs give a real output.

Common mistakes

Including x = 4 in the domain. Division by zero is not allowed.
Setting the numerator equal to zero. Only the denominator matters here.
Forgetting to exclude the value in a written answer.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

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

Show answer

$x \neq 2$
$f(x) = \frac{1}{x + 3}$

Show answer

$x \neq -3$
$f(x) = \sqrt{x - 5}$

Show answer

$x \geq 5$
$f(x) = \sqrt{x + 1}$

Show answer

$x \geq -1$
$f(x) = \frac{1}{x^2 - 9}$

Show answer

$x \neq 3,\ x \neq -3$