How to Find the Domain of a Fraction Function

Problem

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

Answer

x \neq -5

A fraction is undefined wherever its denominator is zero. To find the domain, solve denominator equals zero and remove those values from the real numbers.

Step-by-step solution

Write the rule for a fraction.
$\text{denominator} \neq 0$
Why: Division by zero is undefined, so the denominator cannot be zero.
Set the denominator equal to zero to find the forbidden input.
$x + 5 = 0 \Rightarrow x = -5$
Why: This is the single value that makes the function undefined.
Write the domain as every other real number.
$x \neq -5$
Why: All other inputs give a real output.

Common mistakes

Setting the numerator equal to zero. Only the denominator causes division by zero.
Writing x > -5 or another inequality. A single excluded value leaves the rest of the line open on both sides.
Including x = -5 in the domain.

Practice problems

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

$f(x) = \frac{1}{x - 6}$

Show answer

$x \neq 6$
$f(x) = \frac{1}{2x + 4}$

Show answer

$x \neq -2$
$f(x) = \frac{x}{x - 9}$

Show answer

$x \neq 9$
$f(x) = \frac{1}{x^2 - 16}$

Show answer

$x \neq 4,\ x \neq -4$
$f(x) = \frac{1}{7 - x}$

Show answer

$x \neq 7$