As you can see by the picture the rectangle has length 2x
(the horizontal measurement) and width y (the vertical
measurement).
Area = length·width
Area = (2x)·y
But since y =
,
Area = (2x)·
Area = 2x
And now since we have area expressed as a function of x, we can write
A(x) = 2x
The domain of 2x
is the set of all values of x for
which 2x
is a real number, which will be whenever
4 - x² > 0
(2 - x)(2 + x) > 0
Critical numbers are 2 and -2, so we have to get test points in
the intervals (
, -2), (-2, 2), (2,
). We find
that the only interval in which 4 - x² is non-negative is (-2, 2).
We must exclude the endpoints -2, and +2 because the function would be 0
there, and no rectangle can have 0 area (unless we allow that a horizontal
line segment could be called "a rectangle with width 0" or that a
vertical line segment could be called "a rectangle with length 0"). So the
domain is
(-2, 2)
Edwin