How does one prove or verify that a function is invertible?

Question

Verify that if $f (x) = 5x + x^3 + 2x^5$ is invertible on $\mathbb{R}$, then $f^{-1}$ is differentiable on the same set, and compute $(f^{-1})'(0)$ and $(f^{-1})'(8)$.

Answer 1

There is a general formula $$(f^{-1})'(y)={1\over f'(x)}, \qquad y=f(x)$$ Apply this formula to $x=y=0$ and to $x=1,\ y=8.$

The formula in the Leibniz notation takes the form $${dx\over dy}=\left ({dy\over dx}\right )^{-1}, \quad y=f(x)$$ where the left hand side is evaluated at $y$ and the right hand side at $x.$

Answer 2

Depends on whether you are wanting to show whether it is invertible on its codomain/target or on its range. This question asks you to show its invertible on its codomain.

To do that you will generally need to show two things.

1.) if f(x)=f(y) then x=y

2.) for every y in the codomain, there is some x in the domain which satisfies f(x) = y

Note that if a function is strictly monotone, then 1 is automatically satisfied. You can determine this by looking at the sign of the derivative. If the derivative is always positive for instance, this means that whenever x < y, f(x) < f(y). This follows from the Mean Value Theorem.