Inverse function of a bivariate function with linear constraint

Question

Suppose we have a bijective function $f: x \mapsto y$ with an inverse $f^{-1}$, then from $y$ we can solve for $x$ with: $$f^{-1}(y)=x.$$ Now suppose that we have a bivariate surjective function $g: (x_1,x_2) \mapsto y$. If I impose a linear constraint on $x_1$ and $x_2$ such that it satisfies: $$f(x)=g(\frac{a}{x},\frac{b}{x})=y.$$ From $y$, is it possible for us to express $x$ in terms of $g(x_1,x_2)$ instead of $f(x)$?

Answer 1

As far as I understand you can say this:

If $g$ is invertible, then $g^{-1}(y) = (x_1,x_2)$. But you're looking for $x$ that satisfies $\frac{a}{x} = x_1$ and $\frac{b}{x}=x_2$. In other words $x=\frac{a}{x_1} = \frac{b}{x_2}$.

Hence if $g^{-1}(y)$ is given, and $a,b$ are given. Then you can find $x$.