Find the Image of $\tanh{x}$

Question

Problem

Find the Image of $\tanh{x}$

Attempt

I know that $\tanh{x} \rightarrow 1$ for $x \rightarrow \infty$ and $\tanh{x} \rightarrow -1$ for $x \rightarrow -\infty$. I know that $\tanh{x}$ is continuous over the real numbers and I know that I have to argue from the intermediate value thereom. However, I do not know how to apply the IVT to show that the image is indeed $(-1,1)$. I now that I have to pick an element $y\in (-1,1)$ and….

This is where I am stuck. I do not see how the argue for the double inclusion for

$$\tanh{((-\infty,\infty))}=((-1,1))$$

Answer 1

Hint: You can use the fact that $\tanh x$ is an increasing function: $$ (\tanh x)'=\frac1{\cosh^2 x}>0. $$