Inverse of reciprocal of a function

Question

Can we calculate the inverse of the reciprocal of a function in terms of the function's inverse?

$$g(x)=\frac{1}{f(x)}$$

Now we need to calculate $g^{-1}(x)$ in terms of $f^{-1}(x)$

Answer 1

$x=f(y)\implies y=f^{-1}(x)$

$f(y)=\cfrac {1}{g(y)}$

$\implies f(y)=\cfrac 1{g\big(f^{-1}(x)\big)}$

$\implies g\big(f^{-1}(x)\big)=\cfrac 1{f(y)}=\cfrac 1x$

$\implies g\big(f^{-1}(\cfrac 1x)\big)=x$

$\implies g^{-1}(x)=f^{-1}(\cfrac 1x)$

Not exactly what you want.