Question

If f(x) = (4x + 3) / (6x - 4), x ≠ 2/3, show that f o f(x) = x, for all x ≠ 2/3. What is the inverse of f?

Answer 1

LHS = fof (x) = f(f(x))
$=f\left(\begin{array}{c}\frac{4x+3}{6x-4}\end{array}\right)\\\\=\frac{4\frac{(4x+3)}{(6x-4)}+3}{6\frac{(4x+3)}{(6x-4)}-4}\\\\=\frac{4(4x+3)+3(6x-4)}{6(4x+3)-4(6x-4)}\\\\=\frac{16x+12+18x-12}{24x+18-24x+16}=\frac{34x}{34}\\\\=x= RHS$
hence, fof(x) = x for all x ≠ 2/3

f(x) = (4x + 3)/(6x - 4)
y = (4x + 3)/(6x - 4)
6xy - 4y = 4x + 3
6xy - 4x = 4y + 3
x(6y - 4) = (4y + 3)
x = (4y + 3)/(6y - 4)

$f^{-1}(x) = \frac{(4x+3)}{(6x-4)}$

hence, inverse of f(x) = f(x)