Question

How do you find the domain and range of `y= 1/(x+4) -1`?

Answer 1

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

Explanation:

`"we can express y as"`

`y=1/(x+4)-(x+4)/(x+4)=(-x-3)/(x+4)`

`"the denominator of y cannot be 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"`

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

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

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

`y(x+4)=-x-3`

`xy+4y=-x-3`

`xy+x=-3-4y`

`x(y+1)=-(3+4y)`

`x=-(3+4y)/(y+1)`

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

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

`(-oo,-1)uu(-1,oo)larrcolor(blue)"in interval notation"`
graph{1/(x+4)-1 [-10, 10, -5, 5]}