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