1.5 Composition of Functions
Suppose
that y is a function of x and that x, in turn, is a function of t.
For
example, let y = 5x - 2 and let x = 3 - 2t. Then, using the principle that equals may
be substituted for each other, we arrive at the conclusion that y = 5( 3 - 2t ) - 2. Simplifying the
equation then yields the fact that y
= 13 - 10t. Thus, y is a
function of t.
Stated
as a general principle, if y is a function of x and x is a function of t, then y is also a function of t.
This
is called the principle of Composition of Functions, and it follows from
the defining characteristic of all functions--that there is only one output
number for any given input number. For
any given input number t, there is
only one output number x, and for any
given input number x, there is only
one output number y, thus it follows
that for any given input number t,
there is only one output number y. Thus y is a function of t .
Stated
in this way, the principle of composition of functions is fairly easy to
understand. Furthermore, we can see how
it involves the principle of substitution of equals for equals.
Now
we need to integrate the principle of composition with the idea of function
notation.
In
many contexts, all input numbers will be denoted by the variable x and all output numbers by the variable
y, using function notation to
distinguish the different functions. In
this context, the two functions in the example above would be written as y = 5x
- 2 and y = 3 - 2x. We could distinguish between the two by naming the first function f and the second function g.
Thus we could write f ( x ) = 5x - 2 and
g( x ) = 3 - 2x. How would the principle of composition apply in the context of
function notation?
In
the original example, we put the output of the second function into the first
function. That's exactly what we will
do now. The only difference now is that
the output of the second function is the number g(x). So we want to put g(x) into f(x). That is, we want to find f(g(x) ) .
Now this is where many students become confused. What does f( g(x) ) mean? Here is where the idea of the blank
parenthesis form of a function comes to the rescue. First write the function f in blank parenthesis form: f( )
= 5( ) - 2 . Next, place g(x) inside each blank parenthesis. This gives us f( g(x) ) = 5( g(x)
) - 2 . Then we use the
principle of substitution of equals for equals to replace the g(x) on the right side of the equation with 3 - 2x.
This gives us the result f( g(x)
)
= 5( 3 - 2x) - 2 = 13 - 10x.
Now f ( g(x)
)
is the composition of the functions f
and g, and there is a special symbol
'o' called the composition
symbol to indicate the composition operation.
Definition: (f
o g)(x) = f ( g(x) ).
For
the following exercises, find and simplify the formula for (f o g)(x).
Exercise
1.5.1
, 
Exercise
1.5.2
, 
Exercise
1.5.3
, 
Recall
that the domain of a function is the set of all input numbers and the range is the
set of all output numbers. Thus the
domain of (f o g) will always be a subset of the domain of g. No number can be an
input number for (f o g) unless it is first an
input number of g. In exercise 1.5.2, the domain of g is the set of all real numbers. But in exercise 1.5.3, the domain of g is the set of all non-negative real
numbers. So for exercise 1.5.2, (f o g)(x) = |x| for every real
number x, whereas in exercise 1.5.3,
(f o g)(x) = x
for x > 0. The domains are different, 
versus 
.
For
the final two exercises, find the
formula for (f o g)(x) and specify the
implicit domain using interval notation.
Exercise
1.5.4
, 
Exercise
1.5.5
, 