1. What is the range of k(x)=-|x|
2.Is y=2 a constant function?
3. x=-3 a function?
4. What is the minimum value of y=x^2-4
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1 Answer

1. What is the range of k(x)=-|x|

Asking for the range is another way of asking for valid Y values (or in this case k(x) values). So the range is (-infinity, 0].

2.Is y=2 a constant function?

Yes.

Constant functions have no variation in the output for any input.

3. x=-3 a function?

No it is not. In order to be a function, there are several properties which must be satisfied, one of which is that for any input, there must be a single output. Often we simplify this by asking "does the function pass a vertical line test?" So if you are to draw a vertical line, would it only cross the function in one place for all values of X? For X=-3, it is a vertical line, so it fails this property of functions because there are an infinite number of solutions (Y values) for this single X value (-3).

4. What is the minimum value of y=x^2-4

The minimum value of parabola's will occur either at the vertex (where its slope is 0) or at its domain boundaries. There are no stated domain restrictions for this function and the slope is 0 at X=0. The first derivative is y'=2x and y'=0 at x=0.

The second derivative is y''=2, indicating it is concave for all values X. So the minimum is x=0 and y=-4.

For a function with a higher power, there may be more than one minimum (each called local minimum) and any place the second derivative is concave up, both to the left and right segments where the first derivative is 0 will be one of these local minimums. You will take your inputs for each relative minimum and any domain limits (for functions which do not have all real numbers for their domain) and compare the Y values of the original function to determine the absolute minimum for that function.
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