Domain and Range from a Function Rule

Quick answer: The domain is the set of permitted inputs; the range is the set of outputs actually produced. For real functions, denominators cannot be zero and even-root radicands must be nonnegative.

Worked example

Find the domain and range of f(x) = √(x − 2).

1. Require x − 2 ≥ 0, giving x ≥ 2.
2. A principal square root is nonnegative, so outputs satisfy y ≥ 0.
3. Every y ≥ 0 is obtained by taking x = y² + 2.
4. Domain is [2, ∞); range is [0, ∞).

Check the result

f(2) = 0 and f(11) = 3; negative outputs never occur.

Common mistake to avoid

Do not assume domain and range are identical. A square root denotes the nonnegative root, not both signs.

Try it yourself

1. Find domain and range of 1/(x − 3). 2. Find domain and range of x² over real inputs.

Answers and reasoning

1. Domain excludes 3; range excludes 0, since a reciprocal is never zero. 2. Domain is all real numbers; range is [0, ∞).

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