Please help me!! :( I can't solve this problem!?

Question

I am learning how to solve range in school, and I am having a lot of difficulty learning how to do so. I have two problems that I have to show work for, but somehow, I keep getting the wrong answers:

radical (4 - x^2) FIND RANGE

I keep getting (- infinity, 2] when the answer is supposed to be [2, infinity)

WHAT AM I DOING WRONG?

I can't get this one either: radical (x^2 + 4). Here the answer is [2, infinity)

Please help

Answer 1

For the first one, I suspect a misprint, because the book's answer is wrong. The correct answer is [0, 2]. Your intuition is almost correct -- the part inside the square root will always be less than or equal to 4, and may be any value less than or equal to 4, however where the part inside the square root is less than zero, the square root will not be less than zero, it will just fail to exist (or become complex, but we assume you want to work with real numbers here). So the only possibilities for the part inside the square root are values in [0, 4], so the only possible values for the function itself are in [0, 2].

For the second, well, consider that for any value greater than or equal to 2, you can make x^2 + 4 equal to its square by choosing appropriate values of x. So the range is all numbers greater than or equal to 2, which is [2, ∞).

Answer 2

I get a range of -2 <= y <= 2 for the 1st, and 0 <= y for the second

Answer 3

x-2

Answer 4

As for showing your work, I suggest at least two parts in each case.

One is showing the range for the part that's inside the radical. Based only on that information, you can then show the range for the whole function as given.

Answer 5

√4 - x²

2 - x

- - - - - - - - -s-

Answer 6

(4 - x²)

so range is all the y(vertical; up & down) values included for a function.

So, did you graph it?