Inverse function of $x \ln (x^2+2)$

Question

What is the inverse function of $f(x)=x \ln (x^2+2)$ ?

Assuming it is invertible, and what is the domain?

Answer 1

First of all, your function is can be inversed, because :

$$f'(x)= \frac{2x^2}{x^2+2} + \ln(x^2+2)>0\forall x \in\mathbb R$$

You got the function :

$$f(x) = x\ln(x^2+2)=\ln(x^2+2)^x$$

Write :

$$y=f(x)$$

Then it will be :

$$y=\ln(x^2+2)^x $$

There is no way of solving this equation with respect to $x$ without using binary search or numerical methods. There is no such function that can be found in terms of standard mathematical functions.

The fact that you cannot find a closed type form for the inverse function $f^{-1}(x)$ does not mean that it does not exist though, as we have already proved that $f(x)$ is invertible.

If you want to have a check on the graph of $f^{-1}(x)$ and how it relates to $f(x)$, take a look here.