Domain Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Domain.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The domain of a function is the complete set of allowable input values for which the function produces a defined, valid output.

The domain is the list of valid "questions" you can ask the function — values outside the domain produce undefined or meaningless answers.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: To find the domain, ask: which inputs cause problems? Exclude values that cause division by zero, square roots of negatives, or logarithms of non-positives.

Common stuck point: Default domain is 'all real numbers where the formula works.'

Sense of Study hint: Ask yourself: what values of x would cause division by zero, a negative under a square root, or a log of zero or negative?

Worked Examples

Example 1

easy
Find the domain of f(x) = \sqrt{x - 3}.

Solution

  1. 1
    The expression under a square root must be non-negative: x - 3 \geq 0.
  2. 2
    Solve: x \geq 3.
  3. 3
    The domain is [3, \infty).

Answer

[3, \infty)
Square root functions require non-negative radicands. Setting the expression inside the root \geq 0 and solving gives the domain.

Example 2

medium
Find the domain of g(x) = \frac{1}{x^2 - 4}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the domain of h(x) = \frac{\sqrt{x + 1}}{x - 5}.

Example 2

hard
Find the domain of f(x) = \ln(9 - x^2).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition