Question

How do you find the domain and range of `f(x)= 2/(x-5)`?

Answer 1

The domain is `x in RR-{5}`. The range is `y in RR-{0}`

Explanation:

As you cannot divide by `0`, the denominator is `!=0`

`x-5!=0`, `=>`, `x!=5`

The domain is `x in RR-{5}`

To find the range, proceed as follows :

Let `y=2/(x-5)`

Rearranging

`y(x-5)=2`

`yx-5y=2`

`yx=2+5y`

`x=(2+5y)/y`

Here,

`y!=0`

The range is `y in RR-{0}`

graph{2/(x-5) [-18.02, 18.03, -9.01, 9.01]}

Answer 2

`x inRR,x!=5`
`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-5=0rArrx=5larrcolor(red)"excluded value"`

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

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

`y=2/(x-5)`

`rArry(x-5)=2`

`rArrxy-5y=2`

`rArrxy=2+5y`

`rArrx=(5+2y)/y`

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

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