Question

How do you find the domain and range of `sqrt(x^2- 4)`?

Answer 1

The domain: `(-oo,-2]uuu[2,oo)`
the range: `[0,oo)`

Explanation:

`f(x)=sqrt(x^2-4)`
The best and fastest way is to learn how do parental functions look like and how does the formula look like and then use it.
The domain: square root must be greater or equal to 0:

`x^2-4>=0quad=>quad(x-2)(x+2)>=0`

That happens only when f(x)>0......everything that is above x axis, like this:(i usually draw a simple picture of quadratic function)
`x in (-oo,-2]uuu[2,oo)` (everything that is red)

There are many ways to find the range. I do it like this: parental square root starts in `(x_0,y_0)=>(0,0)` and goes up and curving to the positive values, like this: graph{sqrt(x) [-10, 10, -5, 5]}
As you can see. It is always positive. so the range is: `[0,oo)`