Question

Let $f :R-\left\{\frac{5}{4}\right\}\to R$ be a function defined as $f \left(x\right) = \frac{5x}{4x +5}$ The inverse of $f$ is the map
$g:$ Range $f \to R -\left\{\frac{5}{4}\right\}$ gives by $f^{-1}\left(x\right) =\frac{ax}{a by}$ then the value of $a +b$ is

Answer 1

Given $f\left(x\right)=\frac{5x}{4x +5}; x\in R -\left\{\frac{5}{4}\right\}$
Let $f \left(x\right)=y \Rightarrow x=f^{-1}\left(y\right)$
$\therefore y=\frac{5x}{4x +5}$
$\Rightarrow 4xy + 5y =5x$
$5y =5x -4xy =x\left(5-4y\right)$
$\Rightarrow x =\frac{5y}{5 -4y}$
$g \left(y\right) =f^{-1} \left(y\right) = \frac{5y}{5 -4y} ; R-\left\{\frac{5}{4}\right\}$
or $g \left(x\right) =f^{-1}\left(x\right) =\frac{5x}{5 -4x}$
$\Rightarrow \frac{ax}{a +bx} = \frac{5x}{5 -4x}$
$\Rightarrow a =5, b = -4$
Hence, $a + = 5 - 4 =1$