Verifying f and g are Inverses

Pre-Calculus · Axiom Academy

EXAMPLE Verifying f and g are Inverses Prove two functions are inverses by composing them both ways Verify that f(x) = 5x + 2 and are inverse functions. To prove it, we must show that both f(g(x)) = x and g(f(x)) = x . Inverses undo each other: run a value through one function, then the other, and you land back on the original x — in either order. Nice work — you proved f(x)=5x+2 and are inverses by composing them in both directions. Both directions are required: a pair is inverse only when f(g(x)) = x and g(f(x)) = x . Checking just one is not enough. Substitute, then simplify: drop one function's rule in wherever the other's input variable appears, then cancel. Result: here both compositions collapse to x , so f^ -1 = g and g^ -1 = f . Composing both ways is the definition of inverse functions — each one exactly undoes the other.

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