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

1 Answer
Dec 13, 2017

Answer:

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)enter image source here
#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)#