Question

What is the domain and range of `y= 1/(x-7) -3`?

Answer 1

`x inRR,x!=7`
`y inRR,y!=-3`

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-7=0rArrx=7larrcolor(red)"excluded value"`

`rArr"domain is "x inRR,x!=7`

`(-oo,-7)uu(-7,+oo)larrcolor(blue)"in interval notation"`

`"divide numerator/denominator of "1/(x-7)" by x"`

`y=(1/x)/(x/x-7/x)-3=(1/x)/(1-7/x)-3`

`"as "xto+-oo,yto0/(1-0)-3`

`rArry=-3larrcolor(red)"excluded value"`

`"range is "y inRR,y!=-3`

`(-oo,-3)uu(-3,+oo)larrcolor(blue)"in interval notation"`
graph{1/(x-7)-3 [-10, 10, -5, 5]}