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# End behavior of functions & their graphs

Video transcript

Which function increases as
x increases toward infinity and decreases as x decreases
toward negative infinity? So let's think about each
of these constraints. So, first, which function
increases as x increases? So as x increases
toward infinity. So x is going in that direction. So first let's look
at f of x over here. So f of x, as we get beyond this
minimum point right over here, as we increase our x, f of
x seems to be increasing. So f of x seems to make
the first constraint. Now let's think about g of x. Once we get past this minimum
point right over here, as x gets larger and
larger and larger, as it approaches
infinity, g of x seems to be getting larger
and larger and larger. g of x is moving up. So g of x also seems to
make this first constraint. Now let's think about h of x. As x moves towards
infinity, as x moves towards positive infinity,
h of x seems to be decreasing. So h of x does not even
make the first constraint. So our only two possibilities
are now g of x and f of x. So which of these
decrease as x decreases toward negative infinity? So let's think about
that, x decreasing toward negative infinity. We're going to be going
in that direction. So first let's look at f of x. So f of x, it kind of
goes up and down here. But after we hit this
little local maximum point-- this was a local
minimum point over here, not at a global one--
as we move to the left of this local maximum point,
as we get smaller and smaller x's, we see that the
function is decreasing. So it does seem to meet
the second constraint. It decreases as x decreases
toward negative infinity. So it meets that constraint. Now, what about g of x? After we have this
minimum point-- and actually it looks like
a global minimum point-- after we hit this minimum point
right over here, as x decreases toward negative infinity, g
of x seems to be increasing, not decreasing. So g of x does not meet
the second constraint. So the only function that
met both constraints seems to be f of x.