Question:
Let $f: R-\{-1\} \rightarrow R-\{1\}$ be given by $f(x)=\frac{x}{x+1} .$ Write $f^{-1}(x)$.
Solution:
Let $f^{-1}(x)=y$ $\ldots(1)$
$\Rightarrow f(y)=x$
$\Rightarrow \frac{y}{y+1}=x$
$\Rightarrow y=x y+x$
$\Rightarrow y-x y=x$
$\Rightarrow y(1-x)=x$
$\Rightarrow y=\frac{x}{1-x}$
$\Rightarrow f^{-1}(x)=\frac{x}{1-x}$ $[$ from (1) $]$
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