Prove that $f(x)=5x^3+9$ has inverse in all $\mathbb{R}$.

Question

I have to prove that the function $f(x)=5x^3+9$ has inverse in all $\mathbb{R}$ (I know that graphically it is trivial).

Is it enough to consider that it is strictly monotonic? Is it necessary to prove that it is also continuous?

Thanks in advance!

Answer 1

HINT

Note that

• $f(x)=5x^3+9 \implies f'(x)=15x^2\ge 0 \quad f(x)=0 \iff x=0$

and

• $\lim_{x\to \pm \infty} f(x)=\pm \infty$

what can we conclude about injectivity and surjectivity from here for $f:\mathbb{R}\to \mathbb{R}$?

Answer 2

You don't need to prove that the inverse is continuous indeed there are noncontinuous functions with noncontinuous inverses. Note however that a continuous function if it has an inverse must have a continuous inverse. Write $y = 5x^3+9$ and try to solve for $x$ in terms of $y$. HINT: $x^{3}$ has the inverse $x^{1/3}$

You can also use that $f$ is strictly monotonic. Can you show this?

Answer 3

Continuity has nothing to do with having an inverse. What you want to show is that your function is bijective. That is, $f$ is one-to-one and onto. Also, you could exchange $x$ and $y$ in your function and then solve for $y$ (or just solve for $y$ in terms of $x$).

Edit

It isn't completely true that continuity as nothing to do with having an inverse. While it is not at all necessarily that a continuous function be invertible or invertible function be continuous, in the case where $f$ is continuous and monotonic it is indeed invertible on its range.