PrecalculusPrecalculusHigh school

Domain and Range

Find allowed inputs and possible outputs of a function.

The domain of a function is the set of inputs it accepts; the range is the set of outputs it actually produces. When a function is given by a formula with no stated restriction, the domain is every real number that does not break the formula.

Only a few things break formulas: dividing by zero, taking an even root of a negative number, and taking the logarithm of zero or a negative number. Range is often found from the graph or by solving $y = f(x)$ for $x$ and asking which $y$ values give a valid $x$ .

Key ideas

Denominator: exclude values that make it zero.
Square root: the inside must be $\ge 0$ .
Logarithm: the inside must be $> 0$ .
Write answers in interval notation, using round brackets for excluded endpoints.

Worked example

Find the domain of $f(x) = \frac{\sqrt{x + 3}}{x - 2}$ .

Root: $x + 3 \ge 0$ , so $x \ge -3$ .
Denominator: $x \ne 2$ .
Combine: $x \ge -3$ and $x \ne 2$ .
Interval notation: $[-3, 2) \cup (2, \infty)$ .

Answer: $[-3, 2) \cup (2, \infty)$

Common mistakes

Excluding values that make the numerator zero.
Using a strict inequality for a square root, which wrongly drops the endpoint.
Confusing the range of $x^2$ (all $y \ge 0$ ) with its domain (all reals).

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.What is the smallest $x$ in the domain of $\sqrt{2x - 8}$ ?
Show answer
$4$
2.What value is excluded from the domain of $\frac{5}{3x + 6}$ ?
Show answer
$-2$
3.What is the minimum value in the range of $f(x) = x^2 + 4$ ?
Show answer
$4$

Keep practising

Calculators

Domain and Range Calculator