If f(x)=1 1-x , show that f [f f(x) ]=x - Vidyadip

Question

If $\text{f(x)}=\frac{1}{1-\text{x}},$ show that $\text{f}\big[\text{f}\{\text{f(x)}\}\big]=\text{x}$

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Answer

We have,
$\text{f(x)}=\frac{1}{1-\text{x}}$
Now, $\text{f}\{\text{f}(\text{x})\}=\text{f}\Big\{\frac{1}{1-\text{x}}\Big\}$
$=\frac{1}{1-\frac{1}{1-\text{x}}}$
$=\frac{1}{\frac{1-\text{x}-1}{1-\text{x}}}$
$=\frac{1-\text{x}}{-\text{x}}$
$=\frac{\text{x}-1}{\text{x}}$
$\therefore\ \text{f}\big[\text{f}\{\text{f(x)}\}\big]=\text{f}\Big\{\frac{1}{1-\text{x}}\Big\}$
$=\frac{1}{1-\big(\frac{\text{x}-1}{\text{x}}\big)}$
$=\frac{1}{\frac{\text{x}-\text{x}+1}{\text{x}}}$
$=\frac{\text{x}}{1}$
$=\text{x}$
$\therefore\ \text{f}\big[\text{f}\{\text{f(x)}\}\big]=\text{x}$ Hence, proved.

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