Question

. For which numbers a, b, c, and d will the function?

f(x)=(ax+b)/(cx+d)
satisfy f(f(x))=x for all x?

Answer 1

$f(x)=\dfrac{ax+b}{cx+d}$
$\implies f(f(x))=\dfrac{a\dfrac{ax+b}{cx+d}+b}{c\dfrac{ax+b}{cx+d}+d}$

$x=\dfrac{a\dfrac{ax+b}{cx+d}+b}{c\dfrac{ax+b}{cx+d}+d}$
$x=\dfrac{a(ax+b)+b(cx+d)}{c(ax+b)+d(cx+d)}$
$x=\dfrac{(a^2+bc)x^2+(a+d)b}{(a+d)cx+(bc+d^2)}$
$x=\dfrac{a^2+bc}{(a+d)c}x-\dfrac{(a^2+bc)(bc+d^2)}{(a+d)^2c^2}+\dfrac{(a+d)b+\frac{(a^2+bc)(bc+d^2)^2}{(a+d)^2c^2}}{(a+d)cx+(bc+d^2)}$

For equality to hold, the coefficient of the linear term must be 1, and both the constant term and numerator of the remainder term must vanish.

This will only happen when $a=c$ and $b=d=0$.