Question

I don't know which is which
y=(x+9)^2-3

I know it is all real numbers, but i don't know if my answer should be
[-3,00)
or
(-9,00)

Can some one explain this?

Answer 1

I do not know what your question is asking for so will describe the function.
This parabola opens up (holds water) because y gets big positive as x gets big positive or negative.
The axis of symmetry, and therefore x location of the vertex, is when the quantity (x+9) is zero. That is when x is -9
For example if you go one space left and right of x = -9, you should get the same y
well if x = -9-1 = -10, then y = 1-3 = -2
and if x = -9 + 1 = -8, then y = 1-3 = -2 sure enough the same.
Now to find the vertex, we already know that (x+9) is zero at x = -9 so we need y there, but it is obviously 0-3 = -3
so the location of the vertex is (-9,-3)

Answer 2

Now about your "all real numbers" comment.
Yes, the domain of x is all real numbers, you can find a y for any old x.
However we can not find any y below the vertex at y = -3 because the whole parabola is above the vertex at (-9,-3)
Therefore the Range of y is y greater than or equal to -3