If y=f(x)

If $y=f(x)=\frac{a x+b}{c x-d}$

$f(y)=\frac{a y+b}{c y-d}$

$=\frac{a\left(\frac{a x+b}{c x-d}\right)+b}{c\left(\frac{a x+b}{c x-d}\right)-d}$$f(y)=\frac{a(a x+b)+b(c x-d)}{\frac{c x-d}{\frac{c(a x+b)-d(c x-d)}{c x-d}}}$

$=\frac{a^{2} x+a b+b c x-d b}{a c x+b c-c d x+d^{2}}$

$f(y)=\frac{\left(a^{2}+b c\right) x+a b-b d}{(a c-c d) x+b c-d^{2}}$