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Question
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
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Solution
\[Let f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{y}{y + 1} = x\]
\[ \Rightarrow y = xy + x\]
\[ \Rightarrow y - xy = x\]
\[ \Rightarrow y\left( 1 - x \right) = x\]
\[ \Rightarrow y = \frac{x}{1 - x}\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{x}{1 - x} [\text{from}\left( 1 \right)]\]
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