Convincing others that the method of finding the inverse function is valid

Suppose you have $y=f(x)$. If you change the subject to $x$ and want to get it alone, what you are doing is to find a function $g$ such that $g(f(x)) = x$. To keep the equality we then see that $g(y) = g(f(x)) = x$.

$g(x)$ is the inverse of $f(x)$ if it satisfies $x=f(g(x))$ and $x=g(f(x))$.

On base of $x=f(g(x))$ we go hunting for $g(x)$.

First we abbreviate $g(x)$ by $y$.

Now let's find $y$ on base of the equation $x=f(y)$.

I had a (seemingly un-mathematical) practical method.

Sketch $ y = f(x) $ on a transparent plastic sheet used for projections, the edges serving as x- and y- axes. Flip the sheet swapping x and y along with rigid curve and see. It is so convincing.. no questions will be asked...

EDIT 1:

... as the operation makes it visibly obvious, so at each point you can verify :

1) x and y are interchanged

2) slope is its inverse now, no sign change $ \dfrac{dy}{dx} \rightarrow \dfrac{dx}{dy} $

3) curvature at any point is invariant except sign change, and it can be explained by differentials

$$ \frac{d^2y/dx^2}{(1+y'^{2})^{3/2}} \rightarrow \frac{-d^2x/dy^2}{(1+x'^{2})^{3/2}} $$

4) even higher order isometric invariants are conserved

4) one more flip and you are back; any double transformation annuls, i.e., inverse of an inverse function gives the starting function.