Question

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

Answer 1

It is given that f(x)=(4x+3)/(6x-4),x≠2/3

(fof)(x)=f(f(x))=f[(4x+3)/(6x-4)]
=4(4x+3)/(6x-4)+3/6(4x+3)/(6x-4)-4
=16x+12+18x-12/24x+18-24x+16
=34x/35
=x.

Therefore,fof(x)=x,for all x≠2/3.
⇒fof=I.

Hence, the given function f is invertible and the inverse of f is f itself.