Domain / range of this function

[zeion]

Homework Statement

I don't remember the exact question right now, but it was something like this:

f(x) = sqrt(4 - x^2)

I needed to give the domain / range of it, and also symmetry I believe.

Homework Equations

The Attempt at a Solution

I know this is a half circle after being graphed, but how can I show the domain and range in a math way?

For domain, I know 4 - x^ must >= 0, so I can solve for it that way.
But what about range?
Can I also just isolate x and then get sqrt with y inside and restrict to real numbers?

 

[Tomer]

Formally, you should say: if "y" is an element in the domains, then there's an x, so that:

\sqrt{4 - x^{2}}= y
That already means that y \geq 0, since the square root always gives non-negative values.
Squaring the equation:
=>
x² = 4 - y²

This equation only has a solution if the right side is positive (or zero). therefore:
4 - y² \geq 0
Check out what inequality you get from that.

Of course you need to combine it with y \geq 0 (squaring equations usually leads to extra solutions) with the inequality you got.

A little less formally, but probably valid - you could use the graph of the function. By finding the absolute maximum and minimum of the function in it's closed range [-2,2] and noting that the function gets any values between them, being continuous in that segment - you can find your range.

 

[zeion]

since the square root always gives non-negative values.

\sqrt4 = \pm 2

 

[Tomer]

\sqrt4 = \pm 2

The square root function is defined to be the positive root of a number, unlike the operation of taking a root from a number.