How to Find the Domain of a Square Root Function

Precalculus · Functions and domains

How to Find the Domain of a Square Root Function

Problem

f(x) = \sqrt{x - 7}

Answer

x \geq 7

A square root only accepts zero and positive inputs. The domain is found by writing that condition as an inequality and solving it.

Step-by-step solution

Write the rule for a square root.
$\text{radicand} \geq 0$
Why: The square root of a negative number is not a real number.
Set the expression inside the root greater than or equal to zero.
$x - 7 \geq 0$
Why: The expression under the root is x - 7.
Solve the inequality.
$x \geq 7$
Why: Add 7 to both sides. Adding does not flip the inequality sign.

Common mistakes

Writing x > 7. The endpoint is allowed because the square root of zero is zero.
Flipping the inequality when adding a number. Only multiplying or dividing by a negative flips it.
Setting the radicand equal to zero and stopping. The domain includes every value above the endpoint too.

Practice problems

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

$f(x) = \sqrt{x - 2}$

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$x \geq 2$
$f(x) = \sqrt{x + 6}$

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$x \geq -6$
$f(x) = \sqrt{3x - 12}$

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$x \geq 4$
$f(x) = \sqrt{8 - x}$

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$x \leq 8$
$f(x) = \sqrt{2x + 10}$

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$x \geq -5$