Question

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

Answer 1

Given f(x) = \({\frac{4x + 3}{6x - 4}}\), x ≠ \({\frac{2}{3}}\)

Let us show fof(x) = x

(fof)(x) = f(f(x))

= f((4x + 3)/(6x – 4)) 

=  (4((4x + 3)/(6x -4)) + 3)/(6((4x +3)/(6x – 4)) – 4)

= \(​​{\frac{16x + 12 + 18x - 12}{24x + 18 - 24x + 16}}\)

= (34x)/(34)

= x

So, fof(x) = x for all x ≠ 2/3

=> fof = 1

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