Domain and range

The domain of a function is every input value (x) for which the function produces a valid output; the range is every output value (y) the function can actually produce. Think of a function as a machine: the domain is what you're allowed to feed in, and the range is everything that can come out.

For many functions the domain is all real numbers, but two restrictions come up constantly: you cannot divide by zero, and you cannot take an even root of a negative number (in real numbers). So f(x) = 1/(x − 2) has domain "all real numbers except x = 2," and f(x) = √x has domain x ≥ 0. Ranges follow from the function's behavior: y = x² outputs only y ≥ 0, so its range is y ≥ 0 even though its domain is all real numbers.

On a graph, the domain is the set of x-values the curve covers when squashed onto the horizontal axis, and the range is the set of y-values it covers on the vertical axis. A parabola opening upward from vertex (1, −4) has domain all reals and range y ≥ −4. Interval notation expresses these cleanly: domain (−∞, ∞), range [−4, ∞).

Domain and range questions appear on the ACT, the Praxis Core math exam, and AP Calculus AB, where the domain of an inverse function equals the range of the original. Expect to find domains from equations (watching for zero denominators and negative radicands) and to read domain and range from graphs.