variable in denominator

**sinewave85** · October 10th 2007, 01:47 PM

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

$<br /> f(x)=\frac {ax+b}{cx +d} \ \mbox {or} \ y=\frac {ax+b}{cx +d} <br />$

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.

**Soroban** · October 10th 2007, 02:07 PM

Hello, sinewave85!

Hope this refreshes your memory . . .

Find the inverse of: . $y\:=\:\frac {ax+b}{cx +d}$

Swap the variables: . $x \:=\:\frac{ay+b}{cy+d}$

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

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

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

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

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

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

**sinewave85** · October 10th 2007, 02:21 PM

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

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