You currently have JavaScript disabled on this browser/device. JavaScript must be enabled in order for this website to function properly.

ZingPath: Composition and Inverses

Finding and Using the Rule of Composition of Two Functions

Searching for

Composition and Inverses

Learn in a way your textbook can't show you.
Explore the full path to learning Composition and Inverses

Lesson Focus

Finding and Using the Rule of Composition of Two Functions


Learning Made Easy

You will calculate the rule of composition of two functions.

Over 1,200 Lessons: Get a Free Trial | Enroll Today

Now You Know

After completing this tutorial, you will be able to complete the following:

  • Determine the rule for the composition of two functions.

Everything You'll Have Covered

Functions are a unifying theme in mathematics, and provide a common language for a variety of mathematical phenomenon. A function is a relationship between two sets that maps each element of the first set to exactly one member in the second set. The first set, which can be thought of as a set of inputs, is called the domain. The range, which is equal to the second set, can be thought of as a set of outputs. The notation is used to indicate that f is a function with domain D and range R.

Functions can be represented in a variety of ways, including Venn diagrams, tables, and equations. Venn diagrams are useful for visualizing functions defined on finite sets. For example, let K = {0,2,4} and N = {6,2,1}. The following diagram describes a function :

The arrows in this diagram indicate that f(0) = 2, f(2) = 1, and f(4) = 6. Tables can also be used to represent functions. If K = {0,2,4}, N = {6,2,1} and are as above, then f is described by the following table:

Finally, equations can be used to represent functions. This is the most frequently used method for describing functions.

Function Composition

Function composition is a method used for combining functions. Let , and note that the range of g is contained in the domain of f. It follows that f can be evaluated at g(x) for any x, and therefore, one may define a function by the following rule:

This function is called the composition of f with g. Relative to this composition, g is called the inner function, and f the outer function. For example, let f and g be the real-valued functions defined below:

The composition (f o g)(x) can be defined because the range of g, all real numbers, coincides with the domain of f. In order to evaluate (f o g)(x) at any real number x, f is applied to g(4):

The rule of composition provides a general formula for (f o g)(x). It can be found by applying f to g(x), where x is an indeterminate:

Tutorial Details

Approximate Time 22 Minutes
Pre-requisite Concepts Students should be able to evaluate functions, understand and apply the property of the identity function, calculate the domain and range of a function, apply the commutative rules of multiplication, and identify types of functions and the identity element of an operation.
Course Algebra-1
Type of Tutorial Skills Application
Key Vocabulary composition, composite function, function