Algebra Practical Problem. help greatly appreciated

A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to enclose a rectangular pen on three sides (with the river providing the fourth side). If x is the length of the side perpendicular to the river, determine the area of the pen as a function of x. What is the domain of this function?

If x is the length of the side perpendicular to the river,
then the lengths of the sides of the pen are x, x, and 500-2x

The area of the pen is x(500 - 2x) = 2x(250-x)

I suppose the domain they want is the one that will make the area positive and the lengths of the sides of the enclosure positive.

Notice that 2x(250-x) is zero when x = 0 and x = 250
So we consider three cases

x < 0:
Then x is negative and 250-x is positive. So 2x(250-x) is negative

0 < x < 250:
Then x is positive and 250-x is positive. So 2x(250-x) is positive

x > 250:
Then x is positive and 250-x is negative. So 2x(250-x) is negative

It follows that the domain is [0, 250]

A(x)=x(500-2x)
Domain:
All set of possible x values