Inverse of a function's integral

The function $g$ is strictly positive. Let the function $f$ be defined as

$$f(x) = \int_0^x g(u) du$$

Is there a way to express $f^{-1}(x)$ in terms of $g$?

Solution 1:

If $g$ takes on both negative and positive values, or is zero on some interval, then $f$ is not invertible, as mentioned in comments.

Assume $g$ is strictly positive (or strictly negative), hence $f^{-1}$ exists and is differentiable by inverse function theorem.

Then $f(f^{-1}(x))=x$, so by differentiating, we get that $f'(f^{-1}(x))(f^{-1})'(x) =1$, i.e, $g(f^{-1}(x))(f^{-1})'(x) = 1$.

Thus we see that $f^{-1}$ satisfies the differential equation $$y' = \frac{1}{g(y)}$$

For some functions $g$ this can be solved exactly (for example, $g(y) = e^y$ or $g(y) = y^2+1$), while for others it cannot be solved exactly (for example $g(y) = e^{y^2}$). Hence, you can get this differential formula but no explicit solution in general.