What you will learn

### Define the composition of functions

### How to find the composition of functions

### How to find the domain of composition of functions

### all kinds of questions about composition of functions

Description

In mathematics, **function composition** is an operation that takes two functions *f* and *g* and produces a function *h* such that *h*(*x*) = *g*(*f*(*x*)). In this operation, the function *g* is applied to the result of applying the function *f* to *x*. That is, the functions *f* : *X* → *Y* and *g* : *Y* → *Z* are **composed** to yield a function that maps *x* in *X* to *g*(*f*(*x*)) in *Z*.

The composition of functions is always associative—a property inherited from the composition of relations.

That is, if *f*, *g*, and *h* are composable, then *f* ∘ (*g* ∘ *h*) = (*f* ∘ *g*) ∘ *h*. Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition *g* ∘ *f* is only meaningful if the codomain of *f* equals the domain of *g*; in a wider sense, it is sufficient that the former be a subset of the latter.

Moreover, it is often convenient to tacitly restrict the domain of *f*, such that *f* produces only values in the domain of *g*. For example, the composition *g* ∘ *f* of the functions *f* : ℝ → (−∞,+9] defined by *f*(*x*) = 9 − *x*2 and *g* : [0,+∞) → ℝ defined by {displaystyle g(x)={sqrt {x}}} can be defined on the interval [−3,+3].

Compositions of two real functions, the absolute value, and a cubic function, in different orders, show a non-commutativity of composition.

The functions *g* and *f* are said to commute with each other if *g* ∘ *f* = *f* ∘ *g*. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, |*x*| + 3 = |*x* + 3| only when *x* ≥ 0. The picture shows another example.

The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (*f* ∘ *g*)−1 = *g*−1∘ *f*−1.

Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno’s formula.

Content

### Introduction

### Composition of functions

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