Solving Inverse Equations

Question

I am having some trouble solving the following question:

Find functions $g$ such that the following relation is true:

$x = g^{-1}(1/y)$.

Is there a general method to solve such inverse function equations?

Thanks.

Answer 1

If $g$ is a function of $x$ and $y$ then you cannot satisfy this equation because one could vary $y$ and thus change the right hand side while holding $x$ constant and thus keeping the left hand side constant. This is impossible. So I am going to assume that you make the following definition:

$$y\equiv g(x)$$

If this is not true, then rewrite your question.

The problem is then

$$x=g^{-1}\left(\frac{1}{g(x)}\right)$$

The way to solve this is to recognize that $g(g^{-1}(x))=x$ and thus, applying $g$ to both sides above, we have

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

or

$$g(x)^2=1$$

with solutions

$$g(x)=1,-1$$

so $g(x)$ is a constant function.

Edit: As is mentioned in the comments, the inverse of a constant function is not a function, so there are no solutions to this problem.