Section 1.4 (part 2)- Compositions of Functions

This section includes forming and solving the compositions of functions,  finding the domain of a composite function, identifying composite functions, and looking into problems in which 
f · g=g · f.


Forming and Solving the Compositions of Functions
The composition of functions is a way of combining two functions. The composition of the function f with g is:


(f · g)(x)=f(g(x))

Example: a) Find (f · g)(x) for f(x)=x-3 and g(x)=x² + 4. Evaluate (f · g)(x) when x=3.

(f · g)(x)=f(g(x))

=f(x²+4)

=(x²+4)-3

=x²+1

(f · g)(3)=(3)²+1=10

b) Find (g · f)(x) for f(x)=x-3 and g(x)=x² + 4. Evaluate (g · f)(x) when x=3.

(g · f)(x)=g(f(x))

=g(x-3)

=(x-3)²+4

=(x²-6x+9)+4

=x²-6x+13

(g · f)(3)=(3)²-6(3)+13=4

*Note that (f · g)(x)≠(g · f)(x)

Finding the Domain of a Composite Function

The domain of f · g is the set of all x in he domain of g such that g(x) is in the domain of f. 

  In the chart above, x is the domain of g, and g(x) is the domain of f.

To determine the domain of a composite function such as (f · g)(x), you must restrict the outputs of g(x) so that they are in the domain of f. The domain of g(x) must also be taken into consideration, because 

only the outputs of g will be put into f.

Example: a) Find the domain of the composition (f · g)(x) for f(x)=1/x and g(x)=x-3.

(f · g)(x)=f(g(x))

=f(x-3)

=1/(x-3)

There is no restriction on the domain of g, so the outputs of g can be any real number. The domain of f 

is restricted to all real numbers besides 0. This means that g(x)=x-3≠0, so x≠3. 

The domain of (f · g)(x) is all real numbers except x=3.

                b) Find the domain of the composition of (f · g)(x) for f(x)=x²+2 and 

(f · g)(x)=f(g(x))

=4-x²+2

=2-x²

It may appear that the domain of this function is all real numbers, but this is not the case. The function g  has a restriction on its domain, and therefore there is a restriction on the composition of the functions.

The domain of g is . So, the domain of (f · g)(x) is  also.

Identifying Composite Functions

It is also important to be able to identify the two functions that make up a composite function. To do this, look for an "inner" and an "outer" function. In a composite function (f · g)(x), the outer function will be the f(x) function and the inner function is the g(x) function.

Example: Express the function  as a composition of two functions. 

inner function: g(x)=3x+1

outer function: f(x)=1/x²

this allows you to write this as a composite of two functions:

Looking into Problems in which (f · g)=(g · f)

There are some instances when (f · g)(x)=(g · f)(x); the composition of these functions is the same in both cases.

Example: Find (f · g)(x) and (g · f)(x) given that f(x)=2x+3 and g(x)=0.5(x-3).

(f · g)(x)=f(g(x))

=f(0.5(x-3))

=2(0.5(x-3))+3

=x-3+3

=x

(g · f)(x)=g(f(x))

=g(2x+3)

=0.5((2x+3)-3)

=0.5(2x)

=x

Note that no matter what the value of x is, (f · g)(x) will always equal (g · f)(x), because the composition of the functions is the same.