How to find $g(x)$, if: $f(x)=(x+1)/x$, and $f(g(x))=x$?

Question

How to find $g(x)$, if: $f(x)=\frac{(x+1)}{x}$, and $f(g(x))=x$?

I know that the answer is that $g(x)=\frac{1}{(x-1)}$

But how to come to that answer remains a mystery to me

Please give me some good advice

Thanks in advance :)

Answer 1

Just do it: $$f(\color{red}{g(x)}) = x \implies \frac{\color{red}{g(x)}+1}{\color{red}{g(x)}} = x \implies 1 + \frac{1}{g(x)} =x \implies \frac{1}{g(x)} = x-1,$$ and inverting both sides gives $g(x) = 1/(x-1).$ The point is: when you have $f(\color{red}{\rm something})$, you go in the expression of $f$ and everywhere there's $x$, you put $\color{red}{\rm something}$. Here, our something is $\color{red}{g(x)}$ (see it as a little block).