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.
$\endgroup$ 03 Answers
$\begingroup$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.
$\endgroup$ 2 $\begingroup$Hint: we get $$\frac{dy}{dx}=\frac{e^x+e^{-x}}{2}>0$$ thus $f(x)$ is monotonously increasing and injective and invertible.
$\endgroup$ 0 $\begingroup$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
