Answer
See proof
Work Step by Step
We are given the functions:
$f(x)=\sqrt{x+5}$
$g(x)=x^2-5, x\geq 0$
a) Verify that the two functions are inverse functions.
Determine $f\circ g$ and $g\circ f$:
$(f\circ g)(x)=f(g(x))=f\left(x^2-5\right)=\sqrt {x^2-5+5}=\sqrt{x^2}=|x|=x$
$(g\circ f)(x)=g(f(x))=g\left(\sqrt{x+5}\right)=(\sqrt{x+5})^2-5=x+5-5=x$
We got:
$(f\circ g)(x)=(g\circ f)(x)=x$,
therefore the two functions are inverse functions.
b) Graph $f$ and $g$ and the line $y=x$.
The graphs of $f$ and $g$ are symmetric with respect to the line $y=x$; therefore $f$ and $g$ are inverse functions.
