1. ## variable in denominator

I can't remember how to work with functions\equations where there is a variable in the denominator, like

$\displaystyle f(x)=\frac {ax+b}{cx +d} \ \mbox {or} \ y=\frac {ax+b}{cx +d}$

specificly, how to get the inverse function after you swap the x and y, but in general just how to get the variable out of the denominator. I know you multiply by an expression equivalent to 1, but I can't remember how. This is driving me nuts. Thanks.

2. Hello, sinewave85!

Hope this refreshes your memory . . .

Find the inverse of: .$\displaystyle y\:=\:\frac {ax+b}{cx +d}$
Swap the variables: .$\displaystyle x \:=\:\frac{ay+b}{cy+d}$

Multiply by $\displaystyle (cy+d)\!:\;\;x(cy+d) \:=\:ay+b$

Expand: .$\displaystyle cxy + dx \:=\:ay + b$

Rearrange terms: .$\displaystyle cxy - ay \:=\:b - dx$

Factor: .$\displaystyle y(cx - a) \:=\:b - dx$

Hence: .$\displaystyle y \:=\:\frac{b-dx}{cx-a}$

. . Therefore: .$\displaystyle f^{-1}(x)\:=\:\frac{b-dx}{cx-a}$

3. That helps! I don't know what I was thinking.