Composition of Functions

Composing two functions means feeding the output of one as the input of the other. This operation, denoted , is fundamental to calculus (especially the Chain Rule) and to computer science, where it models the chaining of procedures.

Quick Reference

Concept	Formula / Rule
Composite function
Domain of
Order matters	in general
Associativity

What Is Composition?

If and , we can define a new function by substituting into :

In general, to form the composite , we start with in the domain of , apply to get , and then apply to that result, obtaining . This requires to lie in the domain of .

Machine diagram showing input x through g then f to produce f(g(x)). — (a) Machine diagram: .

Arrow diagram showing the mapping from x to g(x) to f(g(x)). — (b) Arrow diagram for .

Let and be two functions. The composite function is defined by

for all in the domain of for which lies in the domain of .

The domain of is:

Order matters: and are generally different functions.

Examples

Let and . Find and with their domains.

Solution

. :

Since

for all real

always lies in

. So

. :

is defined only for

, and

always lies in

. So

If , find and its domain.

Solution

The formula simplifies to

, but the domain is not all of

. We need:

: .
: , i.e., , i.e., (always true).

So the second condition imposes no additional restriction, and

Decomposing a Function

Any complex function can often be written as a composition of simpler ones. This skill is essential for the Chain Rule in calculus.

Find and such that where .

Solution

To compute

, we first square

, then take the cosine. So:

Check:

. ✓

Find , , such that where .

Solution

To compute

: first take the absolute value, then add 3, then take the square root.

Check:

. ✓

Composing Three or More Functions

Composition is associative: . You can group them in any order when composing more than two functions.

For example, can be written as where , , .

Frequently Asked Questions

Is composition commutative?

Generally no.

and

are usually different functions. For example, with

and

but

. These are not equal.

How do I find the domain of a composite function?

First, find

. Then, from those

-values, keep only those where

falls inside

. The intersection of these two restrictions is

Why is composition important in calculus?

The Chain Rule in differential calculus says: if

, then

. To apply the Chain Rule, you need to recognize a function as a composition and identify its inner and outer parts.

Can a function be composed with itself?

Yes.

is called the iterate of

. For example, if

, then

. Iterates appear in dynamical systems and the study of fractals.

Quick Reference

What Is Composition?

Examples

Decomposing a Function

Composing Three or More Functions

Frequently Asked Questions

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