Find a formula for $f^{-1}(x).$ Identify the domain and range of $f^{-1}.$ Verify that $f$ and $f^{-1}$ are inverses.

$f(x)=\frac{1}{x-9}$

$f^{-1}(x)=\frac{1}{x}9$

$($Simplify your answer. Use integers or fractions for any numbers in the expression.$)$

Identify the domain of $f^{-1}.$ Select the correct choice below and, if necessary, fill in the answer box$($es$)$ to complete your choice.

A$.$

B$.$

C$.$

D$.$

E$.$ The domain is all real numbers.

Identify the range of $f^{-1}.$ Select the correct choice below and, if necessary, fill in the answer box$($es$)$ to complete your choice.

A$.$

B$.$

C$.$

$y$

D$.$