Solve this


$f(x)=\frac{1}{(1-x)}$ then show that $f[f\{f(x)\}]=x$



Given: $f(x)=\frac{1}{(1-x)}$

Need to prove: f[f{f(x)}] = x

Replacing x by 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,


$\Rightarrow f[f\{f(x)\}]=\frac{1-\frac{1}{1-x}}{-\frac{1}{1-x}}$

$\Rightarrow f[f\{f(x)\}]=\frac{\frac{1-x-1}{1-x}}{\frac{-1}{1-x}}$

$\Rightarrow f[f\{f(x)\}]=\frac{-x}{-1}=x_{\text {[Proved] }}$


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