Is there an inverse function of $f(x) = x^2 + \pi\cos x$?

Question

Is there an inverse function of $f(x) = x^2 + \pi\cos x$?

I don't think there is because of the $x^2$, but I don't know how to prove it.

Answer 1

$$f (-x)=(-x )^2-\pi\cos (-x)= f (x)$$

$f $ is not injective and therefore it is not bijective.

So it does not have an inverse function.

Answer 2

It should be declared over which interval we are considering the function.

Indeed f(x) is not invertible as function from $\mathbb{R}\to\mathbb{R}$ but it is invertible, for example, if we assume a restriction $[a,+\infty)\to[b, +\infty)$ indeed

• $f’(x)=2x-\pi\sin x>0$ for some $x\ge a$ and $f(a)=b$

and thus $f:[a,+\infty)\to[b, +\infty)$ is bijective.