Verifying Inverse Functions

Pre-Calculus · Axiom Academy

LESSON Verifying Inverse Functions A candidate inverse is only confirmed when BOTH round trips — f(g(x)) and g(f(x)) — collapse back to x . Picture f and g as two machines. Inverses must cancel in both orders : send x into g then f , and it must come back to x ; send x into f then g , and it must come back to x too. Watch a value make each round trip and land exactly where it started. Take f(x) = 3x - 6 and . To verify, compose them both ways and simplify. Watch each composition collapse : the multiply-by-3 and divide-by-3 cancel, the +6 and -6 cancel, and only x survives. Because f(g(x)) = x and g(f(x)) = x , the functions f and g are genuine inverses. Either composition alone would not have been enough — the rule demands both. Now test f(x) = x^2 and g(x) = 2x . Run the first composition: doubling then squaring gives 4x^2 , which overshoots x for almost every input. One failed round trip is fatal — you can stop right there. Why you can stop: the definition requires both compositions to equal x . The instant one fails, the "and" is broken, so there is no need to check g(f(x)) — f and g are not inverses. Before grinding through algebra, you can probe with one or two specific inputs. If a single value fails to round-trip home, the pair is already disqualified. Test f(x) = x^2 and : at x = 4 both round trips return 4 , but at x = -2 the trip lands on the wrong number.

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