Question:
If $f(x)=\frac{x+1}{x-1}$ then show that $f\{f(x)\}=x$.
Solution:
Given: $f(x)=\frac{x+1}{x-1}$
Need to prove: f{f(x)} = x
Now replacing x by f(x) we get,
$f\{f(x)\}=\frac{f(x)+1}{f(x)-1}$
$\Rightarrow f\{f(x)\}=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}$
$\Rightarrow f\{f(x)\}=\frac{x+1+x-1}{x+1-x+1}$
$\Rightarrow f\{f(x)\}=\frac{2 x}{2}$
$\Rightarrow f\{f(x)\}=x$ [Proved]
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