Functions — Notation, Domain, Range

ACT Math · Axiom Academy

LESSON Functions — Notation, Domain, Range A function is a machine with a rule: every input has exactly one output. Learn to read f(x) , find which inputs are allowed, and find which outputs actually come out the other side. 1. Function Notation — a Machine With a Rule Think of f as a machine: drop a number in for x , and the rule tells the machine exactly what to do to it before it comes out the other side. f(3) doesn't mean " f times 3 " — it means "run the machine with 3 as the input." 2. Domain — Which Inputs Are Allowed The domain is every x -value you're legally allowed to feed the machine. Most rules accept every real number — but two situations knock values OUT of the domain: If x sits in a denominator, whatever makes that denominator 0 is banned — dividing by zero is undefined. and aren't real numbers, so anything that makes the radicand negative is banned. (An ODD root, like , has no such restriction.) 3. Range — Which Outputs Actually Come Out The range is a completely different question from the domain: not "what can I put in," but "what actually comes out the other side." Sweep x across the whole domain and watch every value the machine actually produces — that swept-out set of y -values IS the range. A square is never negative, so the smallest output is 2 (at x=1 ) and it climbs from there. This ratio gets arbitrarily large or small but can never land exactly on 0 . 4. Composition — Feeding One Machine Into Another

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