# Find the domain , range and inverse of f(x) = ax+b / cx+d where a,b,c,d in mathbb(R) and ad-bc =1

Find the domain , range and inverse of f(x) = ax+b / cx+d where $a,b,c,d\in \mathbb{R}$ and ad-bc =1
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juicilysv
Find the domain , range and inverseof f(x) = ax+b / cx+d where a,b,c,d and ad-bc =1domain is R-{cx+d=0} that is $R-\left\{\frac{-d}{c}\right\}$
For inverse of $y=f\left(x\right)=\frac{ax+b}{cx+d}⇒ycx+yd=ax+b⇒cyx-ax=-dy+b$
$⇒x\left(cy-a\right)=-dy+b$
$\left(cy-a\right)=-dy+bx=\frac{-dy+b}{cy-a}$
$⇒x=\frac{-dy+b}{cy-a}$
$⇒{f}^{-1}\left(x\right)=\frac{-dx+b}{cx-a}$
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Paxton Hoffman
The inverse of the finction is $y=\frac{dx-b}{a-cx}$. To find this, u replace all of the x's with y's and all of the y's with x's. then u solve for y.
I think that the domain is all real numbers except -d/c and the range is all of the real numbers except f(-d/c)