Question

What is the range of the function `(x-1)/(x-4)`?

Answer 1

The range of `(x-1)/(x-4)` is `RR"\"{1}` a.k.a. `(-oo, 1) uu (1, oo)`

Explanation:

Let:

`y = (x-1)/(x-4) = (x-4+3)/(x-4) = 1+3/(x-4)`

Then:

`y - 1 = 3/(x-4)`

Hence:

`x-4 = 3/(y-1)`

Adding `4` to both sides, we get:

`x = 4+3/(y-1)`

All these steps are reversible, except division by `(y-1)`, which is reversible unless `y=1`.

So given any value of `y` apart from `1`, there is a value of `x` such that:

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

That is, the range of `(x-1)/(x-4)` is `RR"\"{1}` a.k.a. `(-oo, 1) uu (1, oo)`

Here's the graph of our function with its horizontal asymptote `y=1`

graph{(y-(x-1)/(x-4))(y-1) = 0 [-5.67, 14.33, -4.64, 5.36]}

If the graphing tool allowed, I would also plot the vertical asymptote `x=4`

Answer 2

`y inRR,y!=1`

Explanation:

`"rearrange "y=(x-1)/(x-4)" making x the subject"`

`rArry(x-4)=x-1larrcolor(blue)" cross-multiplying"`

`rArrxy-4y=x-1`

`rArrxy-x=-1+4y`

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

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

`"the denominator of x cannot be zero as this would make"`
`"x undefined."`

`"equating the denominator to zero and solving gives the"`
`"value that y cannot be"`

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

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