Proving that a function is invertible

Question

Let $f: \mathbb R \to \mathbb R $ where $f(x)=\dfrac{e^x - e^{-x}}{2}$ . Prove that $f$ is invertible.

Attempt:

To prove that a function is invertible we need to prove that it is bijective.

The slope at any point is $\dfrac {dy }{dx}= \dfrac{e^x+e^{-x}}{2}$

Now does it alone imply that the function is bijective? How do I proceed from here? I am unable to write the proof formally.

Answer 1

We need to show that

• f is injective and it is guaranteed if f is strictly increasing

• f is surjective and for that we can use limits to $\pm \infty$ and IVT

Answer 2

Hint: we get $$\frac{dy}{dx}=\frac{e^x+e^{-x}}{2}>0$$ thus $f(x)$ is monotonously increasing and injective and invertible.

Answer 3

If $f$ is your function, then $f'(x)=\frac{e^x+e^{-x}}2>0$. So, $f$ is strictly increasing and therefore injective.

And, since $\lim_{x\to\pm\infty}f(x)=\pm\infty$, it follows from the intermediate value theorem that $f$ is surjective.