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Function foundations

Functions, Domain, and Range

Define functions precisely, find valid inputs from common restrictions, and identify range from equations, tables, and graphs without mixing sets.

How this page is maintained

Written for learners, checked against the sources below, and reviewed every year. Last reviewed July 22, 2026.

Short answer

A function assigns each input exactly one output. The domain is the set of allowed inputs, and the range is the set of resulting outputs. Restrictions usually come from zero denominators or even roots of negative expressions.

  • Function means one output per input, not necessarily one-to-one.
  • Domain comes first: check undefined operations before graphing.
  • Range can be found from structure, graph features, or solving for x in terms of y.

Determine whether a relation is a function

From ordered pairs, a relation is a function if no x value maps to two different y values. On a graph, the vertical line test checks the same idea: any vertical line should hit the graph at most once.

Piecewise definitions are still functions if each input belongs to one applicable rule and yields one output.

Find domain restrictions systematically

For rational functions, denominator cannot be zero. For even roots, the inside expression must be nonnegative in real-number settings. For logarithms, the argument must be positive.

  • Rational: denominator not equal to zero.
  • Square root: radicand greater than or equal to zero.
  • Logarithm: argument greater than zero.

Find domain and range of a rational function

Analyze f(x) = (x + 1) / (x - 3) for valid inputs and possible outputs.

  1. Set denominator not equal to zero: x - 3 != 0, so x != 3.
  2. For range, set y = (x + 1)/(x - 3) and solve for x.
  3. Rearrange: yx - 3y = x + 1, so x(y - 1) = 3y + 1.
  4. x = (3y + 1)/(y - 1), so y != 1.
Result: Domain is all real numbers except 3, range is all real numbers except 1.

Common mistakes

  • Assuming every relation in a table is a function without checking repeated x values.
  • Forgetting to exclude denominator zeros from domain.
  • Claiming square roots allow negative radicands in real numbers.
  • Confusing domain and range in interval notation.

Try one

What is the domain of g(x) = sqrt(2x - 8)?

Require 2x - 8 >= 0, so x >= 4. Domain is [4, infinity).

Sources

Power rulesExponents and radicalsSecond-degree equationsQuadraticsExpression structureFactoring

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