Find the inverse of the function: $f(x)=x^9+x$

Question

Let $f(x)=x^9+x$. Show that $f$ has an inverse and find the inverse.

I don't seem to be able to find a way to start tackling this equation. Appreciate any tips on this question.

Answer 1

$$f'(x)=9x^8+1>0$$ for all $x$. Therefore it's strictly increasing and thus injective. Moreover, $$\lim_{x\to \infty }f(x)=+\infty \quad \text{and}\quad \lim_{x\to -\infty }f(x)=-\infty .$$

Therefore $f(\mathbb R)=\mathbb R$ by Intermediate value theorem, and thus it's surjective. Therefore it's bijective.