Function Composition

The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:

$(f∘g)(x)=f(g(x))$

We read the left-hand side as "$f$" composed with $g$ at $x$, and the right-hand side as  "$f$ of $g$ of $x$."  The two sides of the equation have the same mathematical meaning and are equal.  The open circle symbol, $∘$, is called the composition operator. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. 

Function Composition and Evaluation

It is important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. 

In general, $(f∘g)$ and $(g∘f)$ are different functions. In other words, in many cases $f(g(x))≠g(f(x))$ for all $x$. 

Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

Evaluating Composite Functions Using Input Values

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

Example 1 

If $f(x)=-2x$ and $g(x)=x^2-1$, evaluate $f(g(3))$ and $g(f(3))$.

To evaluate $f(g(3))$, first substitute, or input the value of $3$ into $g(x)$ and find the output.  Then substitute that value into the $f(x)$ function, and simplify:

$g(3)=(3)^2-1=9-1=8$

$f(8)=-2(8)=-16$

Therefore, $f(g(3))=-16$

To evaluate $g(f(3))$, find $f(3)$ and then use that output value as the input value into the $g(x)$ function:

$f(3)=-2(3)=-6$

$g(-6)=(-6)^2-1=36-1=35$

Therefore, $g(f(3))=35$

Evaluating Composite Functions Using a Formula

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition $f(g(x))$ or $g(f(x))$. To do this, we will extend our idea of function evaluation. 

In the next example we are given a formula for two composite functions and asked to evaluate the function.  Evaluate the inside function using the input value or variable provided.  Use the resulting output as the input to the outside function.

Example 2 

If $f(x) =-2x$ and $g(x)=x^2-1$, evaluate $f(g(x))$ and $g(f(x))$.

First substitute, or input the function $g(x)$, $x^2-1$ into the $f(x)$ function, and then simplify:

$f(g(x))=-2(x^2-1)$

$f(g(x))=-2x^2+2$

For $g(f(x))$, input the $f(x)$ function, $-2x$ into the $g(x)$ function, and then simplify:

$g(f(x))=(-2x)^2-1 $

$g(f(x))=4x^2-1$

Functional Decomposition

Functional decomposition broadly refers to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition. In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components (which may reflect individual physical processes of interest), or for the purpose of obtaining a compressed representation of the global function; a task which is feasible only when the constituent processes possess a certain level of modularity (i.e., independence or non-interaction).

In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure; i.e. when constituent functions are found to depend on approximately disjointed sets of variables. Also, decomposition of a function into non-interacting components generally permits more economical representations of the function.