Answer
$a)$ $f^{-1}(x)=x^{2}$
$b)$ $f(f^{-1}(x))=f^{-1}(f(x))=x$
Work Step by Step
$f(x)=\sqrt{x}$
$a)$
Substitute $f(x)$ by $y$:
$y=\sqrt{x}$
Interchange $x$ and $y$:
$x=\sqrt{y}$
Solve for $y$:
$x^{2}=(\sqrt{y})^{2}$
$x^{2}=y$
$y=x^{2}$
Substitute $y$ by $f^{-1}(x)$:
$f^{-1}(x)=x^{2}$
$b)$
Verify the equation for the inverse function found by finding $f(f^{-1}(x))$. Substitute $x$ by $f^{1}(x)$ in $f(x)$ and simplify:
$f(f^{-1}(x))=\sqrt{x^{2}}=x$
Complete the verification process by finding $f^{-1}(f(x))$. Substitute $x$ by $f(x)$ in $f^{-1}(x)$ and simplify:
$f^{-1}(f(x))=(\sqrt{x})^{2}=x$
Since $f(f^{-1}(x))=f^{-1}(f(x))=x$, the equation for the inverse function found is correct.
