How do I find the domain and range of $\tanh(x)$

Question

I used the formula $$\frac{e^x - e^{-x}}{e^x + e^{-x}}$$ I calculated the inverse to find the range, but I got the incorrect answer.

Please help me find the domain and range of $\tanh(x)$.

Answer 1

Note that $\tanh x$ is an odd function. So once we know what happens for $x\ge 0$, the rest follows.

We have $\tanh 0=0$. Also, for positive $x$, we have $\tanh x\lt \frac{e^x}{e^x+e^{-x}}\lt 1$. Furthermore, $\lim_{x\to\infty} \frac{e^x-e^{-x}}{e^x+e^{-x}}=1$. Since $\tanh x$ is continuous, it follows by the Intermediate Value Theorem that as $x$ travels over the interval $[0,\infty)$, $\tanh x$ ranges over the interval $[0,1)$.

We leave it to you to find the range of $\tanh x$ as $x$ travels over the interval $(-\infty, 0]$.