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 $f^{-1}\left(x\right)=\frac{ax}{aby}$ then the value of $a+b$ is

Answer 1

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, $a+=5-4=1$