Composite functions are so common that we usually don't think to think
to label them as composite functions. However, they arise any time a
change in one quantity produces a change in another which, in turn,
produces a change in a third quantity. Does that sound confusing?
Don't worry, an example will make things clear.
An Example
For this example, we'll assume that the number of humans living on the
coast affects the number of whales in nearby coastal waters. Since
whales eat plankton, the number of whales will affect the number of
plankton in the waters.
Let's be more specific. Since
whales don't like all the noise that people make, they move out of an
area when too many people move in. If we denote the number of
thousands of people by x and the number of whales by y, a simple model
would be
that
.



Now since the whales are eating the plankton, more whales mean less
plankton. If we measure the amount of plankton by z, then a simple
model is that
.

Now the end result is that the number of people influence the
number of whales which influences the number of plankton. To see how
the number of plankton depend on the number of people, we can compute
that
What's important is that a change is the first quantity, x or the
number of people, produces a change in the second quantity, y or the
number of whales. This, in turn, changes the final quantity, z or the
number of plankton. You can explore this below by moving the red box
on the "People" scale and seeing how the other quantities are influenced.
For your consideration:
 At what rate does the number of whales change compared to the
number of people?
 At what rate does the number of plankton change compared to
the number of whales?
 At what rate does the number of plankton change compared to
the number of people?
 Can you explain the relationship between these three numbers?


Definition
More generally, if we have two functions
and
, we call the new function
the composite of
and
and denote it by
.
Composite functions are much more common than you may realize.
For instance, if you want to compute
on your
handheld calculator, you will enter 1.1 and then press the button
which squares the entry. After that, you will press the button which
exponentiates the entry. Each of the buttons you press, to square and
to exponentiate, represent a function. By feeding what comes out of the
squaring function into the exponentiation function, you are really
computing the composite of these two functions. Said in mathematical
notation, you are first evaluating
at 1.1
and then putting the result into the function
. By composing them, you obtain
which you evaluate at 1.1.

For your consideration:
Express the following functions as composites:




