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# ZingPath: Composition and Inverses

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## Composition and Inverses

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### Lesson Focus

#### Finding and Using the Rule of Composition of Two Functions

Algebra-1

You will calculate the rule of composition of two functions.

### 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