# General Composition of Functions

## The Composition of Two Functions

In the last lecture, we discussed the stretching and shifting of functions.
Stretching and shifting are just two

examples of a general phenomenon called the composition of functions. The
composition of two functions is

the topic for today’s lecture.

Let f(x) and g(x) be two real valued functions whose domains are the real
numbers. Polynomial functions

and sine and cosine functions are examples of functions which fit this
description. Suppose that we took a

number x and applied the function f to it. We would get as our output the number
f(x). Suppose now that

we applied the function g to the number f(x). We would get another number as our
output g(f(x)).

The key point for this section is that we can think of the action of starting
with x, then applying f, then

applying g, not just as two functions being applied one after another, but as
one function, a composition of

f and g. Our new function, which we write as
and speak as “g compose f” or, perhaps more precisely,

as “g after f,” takes every real number x and gives us back the number g(f(x)).

You can think about the composition of functions in the following way:
before, we described a function

as a machine, which takes in raw material, its domain, and gives out a finished
product, its range. Suppose

that we have two such machines, one called f, the other called g. Suppose
further that the raw material for

g is the finished product of f . Then we could imagine setting up these two
machines in an assembly line,

so that as soon as the finished product comes out of the machine f, it goes into
the machine g. How do we

envision the composition
?
What we do is imagine placing a big box around both the machine f and

the machine g, so that when raw material goes into the machine f, we do not see
it again until it comes

out of the machine g. Now, if after we put this big box around f and g, someone
looked at this big box

and saw raw material going in and finished product coming out, he would assume
that this big box was a

single machine, even though we know that in reality it is two machines, working
one after the other. That

person would be perfectly correct , however, to assume that he is seeing only one
machine, because it does

not matter how the machine works inside: all that matters is that it has an
input and an output. Now we,

knowing that this big box encompasses two machines, want to label the box so as
to inform people that this

box contains the machine g working directly after the machine f. We do this by
labelling the big box as

.
It is still a single machine, but now people know how it works inside.

Let us do a few examples of composition using formulas instead of big boxes.
Let f(x) = cos x and let

g(x) = x^3. We get the formula for the composition
by substituting the formula for f(x) in for x in the

formula for g(x). In the case, we substitute cos x for x in g(x) = x^3, so that
we get

Now let us try the composition
the function we get by first applying g and then applying f. This time,

we substitute x^3 in for x in f(x) = cos x. The result is

This example illustrates a very important point about composition: usually,
the function
does not

equal the function
.
The order in which you compose two functions matters a great deal. For another

example, take f(x) = 2x and g(x) = x^2. Then

and

If you were to plot these two compositions alongside the graph of x^2, you
would see that the first,
,
is

a horizontal stretch of the graph of x ^2, while the second,
,
is a vertical stretch of that parabola . This

leads us into the next section.

## Stretches and Shifts as Compositions

For more examples of composition, we study stretches and shifts of functions
again. Remember that we

stated that g(x) is a vertical stretch of f(x) if g(x) = af(x). Let h(x) = ax.
Another way to write the

definition of a vertical stretch is to write g(x) as a composition of f(x) and
h(x). Specifically, g(x) is a

vertical shift of f(x) if g(x) = ()(x)
for some real number a. We see that this is true by finding the

formula for
:

We defined g(x) to be a horizontal stretch of f(x) if g(x) = f(ax) for some
real number a. If we define

h(x) = ax again, then we quickly see that a horizontal stretch can also be
understood to be a composition

of two functions, specifically, g(x) = ()(x),
since when we work out the formula for
,
we get

So, for a vertical stretch, we apply h to the range of f, and for a
horizontal stretch, we apply h to the

domain of f. This should make intuitive sense to you: The range of f corresponds
to the vertical axis on the

xy-plane, and the domain of f corresponds to the horizontal axis. Thus, if you
want to change the graph of

f(x) horizontally, you want to act on its domain, and it you want to change its
graph vertically, you act on

its range.

We said that a function g(x) is a vertical shift of f(x) if for some real
number b we have that g(x) =

f(x)+b. How could we write g(x) as the composition of two functions? One way to
do it is to let k(x) = x+b.

Then

So g is the composition of k after f.

We defined g(x) to be a horizontal shift of f(x) if g(x) = f(x + b) for some
real number b. Clearly g(x)

is the composition of two functions in this case as well: setting k(x) = x + b
again, we see that

Notice how for horizontal shifts k acts on the domain of f, and for vertical
shifts, k acts on the range, just

as in the case of stretches.

## Seeing Functions as Compositions

In the previous section , we took known functions, the stretches and shifts of
f(x), and rewriting them as

the composition of two functions. This is the most important skill that you must
learn about compositions:

recognizing that a function is the composition of two other functions. There are
no simple rules for breaking

up a function into the composition of two other functions; the best way to learn
how this skill is to see a lot

of examples.

First, let us take the example h(x) = sin^{2}x+9 sin x+8. We see that this is a
trigonometric polynomial,

that is, a polynomial of sin x instead of x. This should suggest to you how to
write this function as the

composition of two other functions: the inside function, the one we do first,
should be sin x, and the outside

function should be a polynomial. Specifically, take f(x) = sin x and g(x) = x^{2} +
9x + 8. Then

For the next example, let h(x) = (2x^{2} + 3x - 7)^{4}.
This function is a polynomial raised to the fourth

power. It stands to reason that, to write this function as the composition of
two other functions, g(x) after

f(x), the inside function, f(x), should be the polynomial being raised to the
fourth power , and the outside

function, g(x), should be x^{4}. So let f(x) = 2x^{2} + 3x - 7
and let g(x) = x^{4}. Then

So h(x) is the composition of g(x) after f(x).

Notice the general principle at work here: we look for an inside function, a
function being acted upon,

and we let this function be f(x). We then figure out how f(x) is being acted
upon, and we label the function

that is doing the action by g(x). Let us try two more examples.

Let h(x) = cos(x^{17}). We can see that cos x is
acting on x^{17}. So we let f(x), the inside function, be equal

to x^{17}, and we let g(x), the outside function, be equal to cos x.
Then we get that

so our choices of g(x) and f(x) work.

Finally, take h(x) = sin(cos(x^{2})). This example presents us with a
choice. First, we could take the inside

function to be f(x) = cos(x^{2}) and the outside function to be g(x) =
sin x. Then

so this choice of f(x) and g(x) work. We could also have
chosen the inside function to be x^{2} and the outside

function to be sin(cos x). So let m(x) = x^{2} and n(x) = sin(cos x).
Then we get that

So now we have two equally valid ways to break up h(x)
into the composition of two functions. You may

also notice that we could see h(x) as the action of three different functions ,
one after the other after the

other. In this case, take p(x) = x^{2}, q(x) = cos x, and r(x) = sin x.
Then we define the function to

be a function we get by first acting on x by p, then acting on p(x) by q, and
then acting on q(p(x)) by r.

This triple composition is equal to h(x):

So now we have three different ways to break up h(x) into
the composition of two or more functions. We

will use this skill in the next lecture when we discuss how to differentiate the
composition of functions using

the chain rule .

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