Question

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

Answer 1

It is given that $f\left(x\right)=\frac{(4x+3)}{(6x-4)},x\ne \frac{2}{3}.$
$\left(fof\right)\left(x\right)=f\left(f\right(x\left)\right)=f\left(\frac{(4x+3)}{(6x-4)}\right)$
$=\frac{4\left(\frac{4x+3}{6x-4}\right)+3}{6\left(\frac{4x+3}{6x-4}\right)-4}=\frac{16x+12+18x-12}{24x+18-24x+16}=\frac{34x}{34}=x$
Therefore, fof(x)=x, for all $x\ne \frac{2}{3}\Rightarrow fof=I$
Hence, the given function f is invertible and the inverse of f is itself.