Exercises: (+) Verify by Composition That One Function Is the Inverse of Another

Work through each section in order. For every verification problem, check BOTH compositions: f(g(x)) and g(f(x)).

Grade 9·20 problems·~28 min·Common Core Math - HS Functions·standard·hsf-bf-b-4b

Work through problems with immediate feedback

Fluency Practice

Let $f(x) = x^2$ (all reals) and $g(x) = \sqrt{x}$ . Is $f(g(x)) = x$ for all real $x$ ?

Yes — $(\sqrt{x})^2 = x$ always

No — $f(g(x)) = x$ only for $x \geq 0$ ; $g(x) = \sqrt{x}$ is undefined for $x < 0$

No — the composition gives $|x|$ , not $x$

Yes — both $f(g(x)) = x$ and $g(f(x)) = x$ hold everywhere