The inverse function of $\ln(x + 1)$

Question

The following problems says:

Given the function $ f : A \rightarrow B$, determine the sets $A$ and $B$ for which $ y = \ln(x + 1)$ has an inverse.

How can I solve it, and what are the key aspects to analyze ?

Answer 1

$f $ is defined if $A=(-1,+\infty) .$

$$f'(x)=\frac {1}{1+x} .$$

$f $ is continuous and strictly increasing at $A $.

$f $ is then a bijection from $A $ to $$f (A)=(\lim_{x\to -1^+}f (x),\lim_{x\to +\infty}f (x))$$ $$=(-\infty,+\infty)=\mathbb R =B. $$

remark

You can take for $A $ any subset $C $ of $(-1,+\infty) $ and $B=f (C) $.

Answer 2

This function is defined in $(-1,+\infty)$

$\ln{(x+1)}=\ln{(y+1)} \Rightarrow e^{\ln{(x+1)}}=e^{\ln{(y+1)}} \Rightarrow x+1=y+1 \Rightarrow x=y$.
Thus $f$ is an injection.

Now $y=\ln{(x+1)} \Rightarrow e^y-1=x$

Therefore $f^{-1}(x)=e^x-1$

The range of $f$ by taking limits is the whole real line.