Question 764130
Write the function the way you mean it:  {{{f(x)=x/(x-1)}}}


For easier use of notation, call the inverse of f to be g.


{{{f(g(x))=x=g(x)/(g(x)-1)}}} and also {{{g(f(x))=g(x/(x-1))=x}}}
The {{{f(g(x))}}} composition will be easier to work with.


{{{x=g(x)/(g(x)-1)}}}
{{{x(g(x)-1)=g(x)}}}
{{{x*g(x)-x=g(x)}}}
{{{x*g(x)-g(x)=x}}}
{{{(x-1)g(x)=x}}}
{{{highlight(g(x)=x/(x-1))}}}, the inverse of f(x).


This will help you understand the justification, if not trying a few values in the two functions:

{{{graph(300,300,-8,8,-8,8,x/(x-1))}}}