Mathematics · Functions

Domain and Range of Real Functions

Reference entry · last updated September 20, 2026

A function assigns to each element of its domain exactly one element of its range. For a real-valued function of a real variable, the domain is the set of real inputs for which the rule produces a real output, and the range is the set of all outputs the function actually attains.

1. First principles and interval notation

A relation is a function when no input maps to two different outputs. The vertical line test checks this on a graph: every vertical line meets the graph at most once. The domain and range are sets of real numbers, written in interval notation [1, 2]:

Inequality	Interval	Meaning
\( a < x < b \)	\( (a, b) \)	Open interval, endpoints excluded
\( a \le x \le b \)	\( [a, b] \)	Closed interval, endpoints included
\( x \ge a \)	\( [a, \infty) \)	Unbounded above
\( x < b \)	\( (-\infty, b) \)	Unbounded below
All real \(x\)	\( (-\infty, \infty) \)	Every real number

2. Natural domain restrictions

When a function is given by a formula with no stated domain, the natural domain is every real number for which the formula is defined. Three operations restrict it:

Division by zero: any denominator must be nonzero.
Even roots: a square root, fourth root, and so on require a nonnegative radicand.
Logarithms: the argument of a real logarithm must be strictly positive.

3. Worked domains

\( f(x) = \dfrac{1}{x - 3} \): denominator zero at \(x = 3\), so domain \( (-\infty, 3) \cup (3, \infty) \).
\( g(x) = \sqrt{5 - x} \): radicand \(5 - x \ge 0\), so domain \( (-\infty, 5] \).
\( h(x) = \ln(x + 2) \): argument positive, \(x + 2 > 0\), so domain \( (-2, \infty) \).
\( k(x) = \dfrac{\sqrt{x}}{x - 4} \): combine \(x \ge 0\) with \(x \neq 4\), giving \( [0, 4) \cup (4, \infty) \).

4. Finding the range

The range is the set of outputs. Common routes:

For a parabola \( f(x) = ax^2 + bx + c \), the vertex gives the extreme value; the range is bounded on one side depending on the sign of \(a\).
For a strictly monotonic function, the range follows from the domain via its one-sided limits.
For \( y = f(x) \), solve for \(x\) in terms of \(y\) and require the resulting expression to be defined for \(x\). The set of \(y\) that works is the range.

Example: for \( f(x) = x^2 \), solving \( x = \pm\sqrt{y} \) requires \( y \ge 0 \), so the range is \( [0, \infty) \).

5. Edge cases

A stated domain that is narrower than the natural domain overrides it.
The range can be a single point, as for a constant function.
Piecewise and absolute value functions often have ranges bounded below or above without being monotonic.

References

^OpenStax, College Algebra 2e, Rice University (Domain and range).
^James Stewart, Lothar Redlin, and Saleem Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage Learning, 2015.