The function $f(x)=\frac{5x+1}{x-6}$ is one$-$to$-$one.

a$.$ Find an equation for $f^{-1}(x),$ the inverse function.

b$.$ Verify that your equation is correct by showing that $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x.$

a$.$ Select the correct choice below and fill in the answer box$($es$)$ to complete your choice.

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

A$.f^{-1}(x)=,$ for all $x$

B$.f^{-1}(x)=,$ for $x$

C$.f^{-1}(x)=,$ for $x$

D$.f^{-1}(x)=,$ for $x$