How do I calculate the inverse function of this function?

Question

I have this function: $$ f(x)=\frac{1+\ln(x)}{1-\ln(x)} $$ And i should calculate $f^{-1}(x)$

I am not really sure how to proceed but I think that the first step would be to have x alone, how do I achieve that?

Answer 1

Note, that when inversing a function you should also determine the set over which the inverse function is defined.

Set $y=f(x)$ and solve for $x$: $$y=\frac{1+\ln x}{1- \ln x} \Rightarrow y(1- \ln x)=1+\ln x \Rightarrow y-1=y(\ln x)+\ln x$$ which gives $$(y-1)=(y+1)\ln x\Rightarrow \ln x=\frac{y-1}{y+1}$$ which gives by exponentiating both sides to the power $e$ $$e^{\ln x}=e^{\frac{y-1}{y+1}}$$ which reduces to $$x=e^{\frac{y-1}{y+1}}$$ Therefore $$f^{-1}(y)=e^{\frac{y-1}{y+1}}$$ Note, also that this holds for all $y \in \mathbb{R}\backslash\{-1\}$.

Answer 2

Let $\displaystyle f(x)=\frac{1+\ln x}{1-\ln x}=\frac y1$

$\displaystyle\implies f^{-1}(y)=x$

Applying Componendo & Dividendo,

$$ \ln x=\dfrac{y-1}{y+1}$$

$\displaystyle\implies x=e^{\left(\dfrac{y-1}{y+1}\right)}$ which is $f^{-1}(y)$