Question

How do you find the domain and range of `f(x)=(x-2)/(x+4)`?

Answer 1

`x inRR,x!=-4,y inRR,y!=1`

Explanation:

`"let " y=(x-2)/(x+4)`

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

`"solve "x+4=0rArrx=-4larrcolor(red)"excluded value"`

`rArr"domain "x inRR,x!=-4`

`x in(-oo,-4)uu(-4,oo)larrcolor(blue)"in interval notation"`

`"to find the range, rearrange y making x the subject"`

`rArry(x+4)=x-2`

`rArrxy+4y-x+2=0`

`rArrx(y-1)=-2-4y`

`rArrx=-(2+4y)/(y-1)`

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

`rArr"range "y inRR,y!=1`

`y in(-oo,1)uu(1,oo)`
graph{(x-2)/(x+4) [-10, 10, -5, 5]}