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- anonymous

Hopefully this will be my last question for the day...

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- anonymous

Hopefully this will be my last question for the day...

- katieb

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- anonymous

Does anyone know how to get the domain and range from the following function?...
\[y=-\sqrt{4-x^{2}}\]

- phi

do you know you are not allowed to take the square root of a negative number?

- anonymous

Yes, I do.

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- phi

any idea what x value will make a negative number inside the square root sign?
or, more to the point, what x keeps the inside positive or zero?
4-x^2 >= 0

- phi

to "solve for x" add +x^2 to both sides
4 - x^2 + x^2 >= 0+x^2

- phi

4>= x^2
2 >= x
and -2 <= x (we flip the sign if we pick the negative square root)
so we need
-2 <= x <= 2
that is the domain

- anonymous

"any idea what x value will make a negative number inside the square root sign? or, more to the point, what x keeps the inside positive or zero?"
I'm sorry, but this statement is a bit confusing. I am not sure what exactly you are asking.

- anonymous

Also, how did you get 4-x^2 >= 0 from
\[y=-\sqrt{4-x^{2}}\]

- phi

you want the "stuff" inside the square root to be 0 or bigger , right ?

- anonymous

I haven't done a problem like this in a while, so I don't really have an idea what I am supposed to be doing. Sorry.

- phi

you start with the idea that you don't want to take the square root of a negative number
(because the answer will be undefined)

- phi

we are allowed to take the square root of zero or any positive number
that means that the "stuff" inside the square root, that is
4-x^2 must be either 0 or bigger than 0
we write that as
4-x^2 >= 0
when we simplify that , we will get -2 <= x <-=2
x is from -2 to +2... if it is out side that domain, we will get a negative number when we do 4-x^2, and then we will be trying to take the square root of that negative number.. that is not allowed

- anonymous

Ah! I see now. Thank you :)
So if that is how you find the domain, how do you find the range?

- phi

I would do sqrt(4-0) = sqr(4) = 2
and sqrt(4-2^2)= sqr(0)= 0
we will get numbers from 0 up to 2

- phi

in other words, I know x goes from
-2 to +2
if we pick an "extreme number" like x=-2
and figure out y, we do
\[ \sqrt{4 - (-2)^2} = \sqrt{4-4}=0\]
I also notice that if x=+2 we will get the same answer for y
then as we pick x's closer to 0, y will get bigger, until we get to x=0
at which point y = sqr(4-0)= 2
so the range will be 0 to 2

- anonymous

So you pick an "extreme number" to put in the x's place, and whatever answer you get for y, you then place it into the x's place to receive the next number?
(my wording may be confusing)

- phi

No, not that
we know the domain is x= -2 to +2
I also know (because we are doing x^2 in sqr(4-x^2)
that the negative x's will give the same y value as the positive x's
so the range will be the y values from x=0 to x=2
so I found the y value for x=0 and the y value for x=2
and I know the y values will be between those two numbers.

- phi

maybe this will be helpful
https://www.khanacademy.org/math/algebra2/functions_and_graphs/domain_range/v/domain-of-a-function

- anonymous

Sorry, I had to step away for a bit.
I do understand what you're saying now. I just needed to re-read it a few times.

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