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