Domain & Range

Domain is the set of inputs a function accepts. Range is the set of outputs it produces.

From a list of pairs, just read them off. For {(1,4), (2,7), (5,7)}: domain {1, 2, 5}, range {4, 7}.

From a formula, the domain is everything except values that break the maths. Two things break it:

• dividing by zero, so exclude any x making a denominator 0
• square-rooting a negative, so require the inside to be ≥ 0

From a graph, scan left-to-right for domain and bottom-to-top for range.

Real contexts add their own limits. If x is a number of people, negative and fractional values are out regardless of what the formula allows.

The domain of a function given by a formula is, by convention, its natural domain: the largest set of reals for which the expression is defined. The two restrictions that arise in Algebra 1 are denominators, requiring ≠ 0, and even-index radicals, requiring the radicand ≥ 0. Later, logarithms add the requirement that the argument be strictly positive.

When both restrictions occur, they intersect. For f(x) = √(x − 2)/(x − 5) the domain is x ≥ 2 and x ≠ 5, which is [2, 5) ∪ (5, ∞).

Range is generally harder because it depends on the function's behaviour rather than a local rule. For a parabola y = a(x − h)² + k the vertex settles it: the range is [k, ∞) when a > 0 and (−∞, k] when a < 0. For other families, transformations of a known parent function are usually the fastest route.

A restricted domain imposed by context — sometimes called the practical domain — can be smaller than the natural one, and applied problems should state it.