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

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

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

$=\frac{16x+12+18x-12}{24x+18-24+16}$

$=\frac{34x}{34}=x.$

$y=\frac{4x+3}{6x-4}$

$6xy-4y=4x+3$

$6xy-4x=4y+3$

$x=\frac{4y+3}{6y-4}$

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