Let f : R − {−1} → R − {1} be given by
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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