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

Question

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

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Answer

We have, $\text{f(x)}=\frac{1+\text{x}}{1-\text{x}}$ Now, $\text{f}\big[\text{f}\text{(x)}\big]=\text{f}\Big(\frac{\text{x}+1}{1-\text{x}}\Big)$ $=\frac{\big(\frac{\text{x}+1}{1-\text{x}}\big)+1}{\big(\frac{\text{x}+1}{1-\text{x}}\big)-1}$ $=\frac{\frac{\text{x}+1+{\text{x}}-1}{{\text{x}}-1}}{\frac{\text{x}+1-1(\text{x}-1)}{\text{x}-1}}$ $=\frac{\frac{2\text{x}}{\text{x}-1}}{\frac{\text{x}+1-\text{x}+1}{\text{x}-1}}$ $=\frac{2\text{x}}{2}$ $=\text{x}$ $\therefore\ \text{f}\big[\text{f}\text{(x)}\big]=\text{x}$ Hence, proved.

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