What is the domain and range of `y =sqrt(x-3) - sqrt(x+3)`?

Given: `y = sqrt(x-3) - sqrt(x + 3)`

Both the domain is the valid inputs `x`. The range is the valid outputs `y`.

Since we have two square roots, the domain and the range will be limited.

`color(blue)"Find the Domain:"`

The terms under each radical must be `>= 0`:

`x - 3 >= 0; " " x + 3 >= 0`

`x >= 3; " "x >= -3`

Since the first expression must be `>=3`, this is what limits the domain.

Domain: `[3, oo) " or " x >= 3`

`color(red)"Find the Range:"`

The range is based on the limited domain.

Let `x = 3 => y = sqrt(3-3) - sqrt(3+3) = -sqrt(6)`

Let `x = 100 => y = sqrt(97) - sqrt(103) ~~-.3`

Let `x = 1000 => y = sqrt(997) - sqrt(1003) ~~-.09`

`x -> oo, y -> 0`

Range: `[-sqrt(6), 0) " or " -sqrt(6) <= y < 0`

What's under the `sqrt` sign must be `>=0`

`=>`, `x-3>=0` and `x+3>=0`

`=>`, `{(x>=3),(x>=-3):}`

Therefore,

The domain is `(x>=3)nn(x>=-3)`

That is, `x in [3,+oo)`

When `x=3`, `=>`, `y=0-sqrt6`

And when `x->+oo`

`lim_(x->+oo)y=0^-`

Therefore,

The range is `y in [-sqrt(6),0^-)`

graph{sqrt(x-3)-sqrt(x+3) [-1.42, 18.58, -6.36, 3.64]}

Given:

`y = sqrt(x-3)-sqrt(x+3)`

First note that the square roots are well defined and real if and only if `x-3 >= 0` and `x+3 >= 0`. Hence it is necessary and sufficient that `x >= 3`.

So the domain of the function is `[3, oo)`

To find the range, note that when `x = 3` then:

`y = sqrt((color(blue)(3))-3)-sqrt((color(blue)(3))+3) = sqrt(0)-sqrt(6) = -sqrt(6)`

We find:

`lim_(x->oo) (sqrt(x-3)-sqrt(x+3)) = lim_(x->oo) ((sqrt(x-3)-sqrt(x+3))(sqrt(x-3)+sqrt(x+3)))/(sqrt(x-3)+sqrt(x+3))`

`color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = lim_(x->oo) ((x-3)-(x+3))/(sqrt(x-3)+sqrt(x+3))`

`color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = lim_(x->oo) (-6)/(sqrt(x-3)+sqrt(x+3))`

`color(white)(lim_(x->oo) (sqrt(x-3)-sqrt(x+3))) = 0`

Note that `-6/(sqrt(x-3)+sqrt(x+3))` is continuous and monotonically increasing.

Hence the range of the given function runs from the minimum value `-sqrt(6)` up to but not including the limit `0`.

That is, the range is `[-sqrt(6), 0)`

graph{y = sqrt(x-3)-sqrt(x+3) [-10, 10, -5, 5]}