Show that the given function has an inverse and hence find its inverse.

Question

Show that the function $f(x) = (e^x - e^{-x})/(e^x + e^{-x})$ has an inverse function for all $x$. Hence find the inverse.

So, I don't know how to show the function has an inverse but if it does then I did this:

$x = (e^y - e^{-y})/(e^y + e^{-y})$

$x = (e^{2y} - 1)/(e^{2y} +1)$

$x (e^{2y} +1) = e^{2y} - 1$

$e^{2y}(1 - x) = (1 + x)$

:. $y = 0.5 ln(1+x) - 0.5ln(1-x)$

Now, how to show it has an inverse for all $x$ without literally showing it's inverse?

Answer 1

Write the function as $$\frac{e^x}{e^x+e^{-x}}-\frac{e^{-x}}{e^x+e^{-x}}=\frac{1}{1+e^{-2x}}-\frac{1}{1+e^{2x}}.$$

Differentiating then gives $$\frac{2e^{-2x}}{(1+e^{-2x})^2}+\frac{2e^{2x}}{(1+e^{2x})^2},$$ which is positive for all $x.$ Thus the function is increasing for all $x,$ hence it is monotonic, hence it is injective, hence it is invertible.