Problem 387 (difficulty: 2/10)
Show that the following functions are injective on the given set \(\displaystyle H\), and calculate the inverse.
(1) \(\displaystyle f(x)=\frac{x}{x+1}, \ H=[-1,1]\); (2) \(\displaystyle f(x)=\frac{x}{|x|+1}, \ H=\R.\)
Solution:
\(\displaystyle f^{-1}(y)=\frac{y}{1-|y|},\ y\in (-1,1)\).