According to this theory, sleep is a product of evolution.
(what is the theory? )

The answer to your question would be restorative theory.

Domain is all real numbers.

The range is

The function is increasing over .

The function is decreasing over

The function has a positive y-intercept.

-----------------This is a guess if I had interpreted your choices correctly:

Second option: The range is

Third option:  The function is increasing over .

Last option:  The function has a positive y-intercept.

I can't really read some of your choices. So you can read my above and determine which is false. If you have a question about any of what I said above please let me know.

Note: I guess those 0's are suppose to be infinities? I hopefully your function is .

Step-by-step explanation:

is a polynomial function which mean it has domain of all real numbers.  All this sentence is really saying is that there exists a number for any value you input into .

Now since the is a quadratic then it is a parabola. We know it is a quadratic because it is comparable to , .

This means the graph sort of looks like a U or an upside down U.

It is U, when .

It is upside down U, when .

So here we have so which means the parabola is an upside down U.  

Let's look at the range.  We know the vertex is either the highest point (if ) or the lowest point (if ).

The vertex here will be the highest point, again since .

The vertex's x-coordinate can be found by evaluating :

.

So the y-coordinate can be found by evaluate for :

So the highest y-coordinate is 6.  The range is therefore .

If you picture the upside down U in your mind and you know the graph is symmetrical about x=-2.

Then you know the parabola is increasing on and decreasing on .

So let's look at the intevals they have:

So on the function is increasing.

Looking on the function is increasing on (-4,-2) but decreasing on the rest of that given interval.

The function's y-intercept can be found by putting 0 in for :

The y-intercept is positive since 2>0.