Composition of Functions

Composition of functions is a fundamental concept in mathematics, especially in the study of algebra and calculus. When you compose two functions, you apply one function to the result of the other function. The composition of functions is denoted by a small circle, such as f • g, where f and g are the functions being composed.

Composition Notation

The composition of functions f and g is denoted as (f • g)(x) or f(g(x)). This means that you apply the function g to the input x, and then apply the function f to the result of g(x).

Example

If f(x) = 2x + 1 and g(x) = x², then the composition (f • g)(x) is given by f(g(x)) = f(x²) = 2(x²) + 1 = 2x² + 1.

Properties of Composition

When working with compositions of functions, it's important to understand some key properties:

1. Associative Property: (f • g) • h = f • (g • h)
2. Identity Property: f • I = I • f = f, where I(x) = x (the identity function)

Study Guide

When studying composition of functions, make sure to focus on the following aspects:

• Understanding the notation of composition and how to represent it mathematically.
• Practicing the process of composing two or more functions step by step.
• Recognizing the properties of composition and how they can be applied in problem-solving.
• Applying compositions of functions to real-world scenarios or practical problems.

By mastering the concept of composition of functions, you'll have a strong foundation for more advanced topics in mathematics, such as function transformations and calculus.