Question

For `f(x) = 1/(x-3)`, what is the natural domain and range?

Answer 1

`x inRR,x!=3`
`y inRR,y!=0`

Explanation:

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

`"solve "x-3=0rArrx=3larrcolor(red)"excluded value"`

`"domain is "x inRR,x!=3`

`(-oo,3)uu(3,+oo)larrcolor(blue)"in interval notation"`

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

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

`rArry(x-3)=1`

`rArrxy-3y=1`

`rArrxy=1+3y`

`rArrx=(1+3y)/y`

`rArry=0larrcolor(red)"excluded value"`

`"range is "y inRR,y!=0`

`(-oo,0)uu(0,+oo)larrcolor(blue)"in interval notation"`
graph{1/(x-3) [-10, 10, -5, 5]}