Question

What is the domain and range of `f(x) = (x-2) / (x+2)`?

Answer 1

`x inRR,x!=-2,y inRR,y!=1`

Explanation:

The denominator of f(x) cannot equal zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be.

`"solve "x+2=0rArrx=-2larrcolor(red)"excluded value"`

`rArr"domain "x inRR,x!=-2`

`x in (-oo,-2)uu(-2,oo)larrcolor(blue)"in interval notation"`

`"let " y=(x-2)/(x+2)`

`"For range rearrange making x the subject"`

`rArry(x+2)=x-2`

`rArrxy+2y=x-2`

`rArrxy-x=-2-2y`

`rArrx(y-1)=-2(1+y)`

`rArrx=-(2(1+y))/(y-1)`

`"solve "y-1=0rArry=1larrcolor(red)"excluded value"`

`"Range "y inRR,y!=1`

`y in(-oo,1)uu(1,oo)`
graph{(x-2)/(x+2) [-10, 10, -5, 5]}