Domain and Range

Pre-Calculus · Axiom Academy

Every function is a machine: the domain is what you're allowed to put in, the range is what can come out. Think of a function as a machine. You feed in an input x , the machine applies its rule, and one output f(x) comes out the other side. The collection of every input you're allowed to feed it is the domain ; the collection of every output it can hand back is the range . Domain = the set of allowed inputs (the x -values) Range = the set of resulting outputs (the y -values) Every input value the function is defined for. Ask: "What can I put IN?" Every output value the function can produce. Ask: "What can come OUT?" Often the domain is "all real numbers" — but two operations are not always legal , and each one carves something out of the number line. The domain is whatever is left over . A denominator can never equal 0 . Exclude every input that makes it zero. A square root needs its inside to be (in the reals). Keep only inputs that satisfy that. A single excluded point leaves a two-piece domain: the domain of is . A one-sided cutoff leaves a ray: the domain of is , where the square bracket means x=2 is included. On a graph you don't have to guess. Flatten the curve onto the x -axis and the shadow it covers is the domain. Flatten it onto the y -axis and that shadow is the range. Domain is the graph's horizontal extent ; range is its vertical extent . Domain (horizontal shadow): the curve spreads across every x , so the domain is all real numbers, .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.