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