Question

How do you find the domain and range of `y = { x/ (x + 5) }`?

Answer 1

`x inRR,x!=-5`
`y inRR,y!=1`

Explanation:

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+5=0rArrx=-5larrcolor(red)" excluded value"`

`rArr"domain is "x inRR,x!=-5`

To find any excluded values in the range rearrange making x the subject.

`rArry(x+5)=xlarrcolor(blue)" cross-multipling"`

`rArrxy+5y=x`

`rArrxy-x=-5ylarrcolor(blue)" collect term in x together"`

`rArrx(y-1)=-5ylarrcolor(blue)" factor out x"`

`rArrx=-5y/(y-1)`

`"the denominator cannot equal zero"`

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

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