Answer
$a)$ $f^{-1}(x)=\dfrac{4}{x-9}$
$b)$ $f(f^{-1}(x))=f^{-1}(f(x))=x$
Work Step by Step
$f(x)=\dfrac{4}{x}+9$
$a)$
Substitute $f(x)$ by $y$:
$y=\dfrac{4}{x}+9$
Interchange $x$ and $y$:
$x=\dfrac{4}{y}+9$
Solve for $y$:
$x-9=\dfrac{4}{y}$
$y(x-9)=4$
$y=\dfrac{4}{x-9}$
Substitute $y$ by $f^{-1}(x)$:
$f^{-1}(x)=\dfrac{4}{x-9}$
$b)$
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:
$f(f^{-1}(x))=\dfrac{4}{\Big(\dfrac{4}{x-9}\Big)}+9=\dfrac{4(x-9)}{4}+9=...$
$...=x-9+9=x$
Complete the verification process by finding $f^{-1}(f(x))$. Substitute $x$ by $f(x)$ in $f^{-1}(x)$ and simplify:
$f^{-1}(f(x))=\dfrac{4}{\dfrac{4}{x}+9-9}=\dfrac{4}{\Big(\dfrac{4}{x}\Big)}=\dfrac{4x}{4}=x$
Since $f(f^{-1}(x))=f^{-1}(f(x))=x$, the equation for the inverse function found is correct.