Question

If $y=f\left(x\right)=\frac{ax+b}{cx-d},\mathrm{then}f\left(y\right)=$ __________ .

Answer 1

$$\begin{gathered} \mathrm{If}y=f\left(x\right)=\frac{ax+b}{cx-d} \\ f\left(y\right)=\frac{ay+b}{cy-d} \\ =\frac{a\left({\displaystyle \frac{ax+b}{cx-d}}\right)+b}{c\left({\displaystyle \frac{ax+b}{cx-d}}\right)-d} \\ f\left(y\right)=\frac{a\left(ax+b\right)+b\left(cx-d\right)}{{\displaystyle \frac{cx-d}{{\displaystyle \frac{c\left(ax+b\right)-d\left(cx-d\right)}{cx-d}}}}} \\ =\frac{a^{2}x+ab+bcx-db}{acx+bc-cdx+d^2} \\ f\left(y\right)=\frac{\left(a^2+bc\right)x+ab-bd}{\left(ac-cd\right)x+bc-d^2} \end{gathered}$$