Verify Functions Are Inverses Practice Test

Verify by composition whether $f(x)=\dfrac{1}{x-1}$ and $g(x)=1+\dfrac{1}{x}$ are inverses. Compute $f(g(x))$ and $g(f(x))$ and simplify (assume inputs avoid values that make denominators zero).

A student claims $f(x)=x^2$ (with domain restricted to $x\ge 0$) and $g(x)=\sqrt{x}$ are inverses. Verify the claim by showing whether both $f(g(x))$ and $g(f(x))$ equal $x$ on the appropriate domains.