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Question
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
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Solution
\[\text{Let }f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{2y}{5y + 3} = x\]
\[ \Rightarrow 2y = 5xy + 3x\]
\[ \Rightarrow 2y - 5xy = 3x\]
\[ \Rightarrow y\left( 2 - 5x \right) = 3x\]
\[ \Rightarrow y = \frac{3x}{2 - 5x}\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{3x}{2 - 5x} [\text{from}\left( 1 \right)]\]
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