Question

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

Answer 1

`x >= 3` or

in interval notation `[3, oo)`

Explanation:

Given: `F(x) = sqrt(x - 3)`

A function starts out having a domain of all Reals `(-oo, oo)`

A square root limits the function because you can't have negative numbers under the square root (they are called imaginary numbers).

This means `" "x - 3 >= 0`

Simplifying: `" "x >= 3`

Answer 2

The domain is `x in [3, +oo)`. The range is `y in [0, +oo)`

Explanation:

Let `y=sqrt(x-3)`

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

Therefore,

`x-3>=0`

`=>`, `x>=3`

The domain is `x in [3, +oo)`

When `x=3`, `y=sqrt(3-3)=0`

And

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

Therefore,

The range is `y in [0, +oo)`

graph{sqrt(x-3) [-12.77, 27.77, -9.9, 10.38]}