How do you find the range of a function algebraically ##y=(x+5)/(x-2)##?

To best way to find the range of a function is to find the domain of the inverse function. To find the inverse function of a function you have to substitue ##x## with ##y##, and vice versa, and then find ##y##.

So:

##y=(x+5)/(x-2)rArrx=(y+5)/(y-2)rArrx(y-2)=y+5rArr##

##xy-2x=y+5rArrxy-y=2x+5rArry(x-1)=2x+5rArr##

##y=(2x+5)/(x-1)## and its domain is ##(-oo,1)uu(1,+oo)## that is, also, the range of your function.