Question

If $f\left(x\right)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$, show that $fof\left(x\right)=x$, for all $x\ne \frac{2}{3}$. What is the inverse of $f$?

Answer 1

$f\left(x\right)=\frac{(4x+3)}{(6x-4)}$

Now, $f\left(f\right(x)=f\left(\frac{(4x+3)}{(6x-4)}\right)=\frac{4\times \frac{(4x+3)}{(6x-4)}+3}{6\times \frac{(4x+3)}{(6x-4)}-4}=\frac{16x+12+18x-12}{24x+18-24x+16}=\frac{34x}{34}=x$
Since,
$f\left(f\right(x\left)\right)=x⟹f^{-1}f\left(f\right(x\left)\right)=f^{-1}\left(x\right)$
$⟹f^{-1}\left(x\right)=f\left(x\right)$
$\therefore f^{-1}\left(x\right)=\frac{(4x+3)}{(6x-4)}$