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

Assuming it is invertible, and what is the domain?

Jaideep Khare
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Lil Read
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  • What did you try so far? – Jaideep Khare Nov 29 '17 at 13:33
  • Welcome to MSE. Here's a MathJax Tutorial !. Posting questions without showing us your effort isn't good, as this is not a "solve my homework" website. The particular question and function though leads to nowhere by trying to calculating the inverse, so that's the reason I gave you an explanation in the answer window below. For your future questions, try to provide us with your attempt at a solution ! – Rebellos Nov 29 '17 at 13:47
  • Also, check if your question is correctly stated. It seems weird to me that they'd give you a non-computable over standard mathematical functions question if you're still into entry mathematics. – Rebellos Nov 29 '17 at 14:11

1 Answers1

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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.

Rebellos
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