Question

What is the domain and range of `y=(4+x)/(1-4x)`?

Answer 1

The domain is `RR-{1/4}`
The range is `RR-{-1/4}`

Explanation:

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

As you cannot divide by `0`, `=>`, `1-4x!=0`

So,

`x!=1/4`

The domain is `RR-{1/4}`

To find the range, we calculate the inverse function `y^-1`

We interchange `x` and `y`

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

We express `y` in terms of `x`

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

`x-4xy=4+y`

`y+4xy=x-4`

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

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

The inverse is `y^-1=(x-4)/(1+4x)`

The range of `y` is `=` to the domain of `y^-1`

`1+4x!=0`

The range is `RR-{-1/4}`

Answer 2

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

Explanation:

`"the domain is defined for all real values of x, except"`
`"those values which make the denominator zero"`

`"to find excluded values equate the denominator to zero"`
`"and solve for x"`

`"solve " 1-4x=0rArrx=1/4larrcolor(red)"excluded value"`

`rArr"domain is " x inRR,x!=1/4`

`"to find any excluded values in the range, change the subject"`
`"of the function to x"`

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

`rArry-4xy=4+x`

`rArr-4xy-x=4-y`

`rArrx(-4y-1)=4-y`

`rArrx=(4-y)/(-4y-1)`

`"the denominator cannot equal zero"`

`rArr-4y-1=0rArry=-1/4larrcolor(red)" excluded value"`

`rArr"range is " y inRR,y!=-1/4`