Question

Any help would be greatly appreciated:

If f(x)=x/(x-1)
Prove f(f(x))=x

In the solution book it shows this progression, but I don't know how the equation is simplified to x/(x-(x-1)) as below.

f(x/(x-1)) = (x/(x-1))/(x/((x-1)-1) = x/((x-(x-1)) = x, provided x not equal to 1

source: Calculus with Analytic Geometry Student Solution book for Calculus with Analytic Geometry 2nd Ed. by George F. Simmons chapter 1.5 question 15 (the concept of a function).

Thanks,
Alan

Answer 1

\[f(x)=\frac{ x }{ x-1 }\]
\[f(f(x))=\frac{ \frac{ x }{ x-1 } }{ (\frac{ x }{ x-1 })-1 }\]
now if we multiply both the top and the bottom with x-1, what will we get?

Answer 2

Try rewrite it as a division:
\[(\frac{ x }{ x-1 })\div(\frac{ x }{ x-1 }-1)\]

Answer 3

Thank you, I really appreciate it.