Chapter 1: Graphs and Functions

1.8 Combining Functions; Composite Functions

This section explores how to combine functions using basic operations and how to form and analyze composite functions. Understanding these concepts is essential for modeling and solving real-world problems in mathematics and science.

Basic Operations on Functions

• Sum:

• Difference:

• Product:

• Quotient: , where

The domain of each operation is the set of all values common to the domains of and , with the additional restriction for the quotient that .

Example: Combining Functions

Let and . Find the following:

• (f - g)(4): , , so

• (f + g)(x):

• (f - g)(x):

• (fg)(x):

Finding the Domain of Combined Functions

To find the domain of , , , and , determine the intersection of the domains of and , excluding values where for the quotient.

For example, if is defined for and for , then the domain of their combination is .

Composite Functions

The composition of with is written as . The domain of consists of all in the domain of such that is in the domain of .

Example: Evaluating Composite Functions

• Let and .

Finding the Domain of a Composite Function

To find the domain of :

1. Find the domain of (call it ).

2. Find the set of in for which is in the domain of .

3. The domain of is .

For example, if is defined for and is defined for , then is defined for those such that .

Decomposing a Function

Given a function , you can write it as a composition by identifying an inner function and an outer function such that .

• Example: can be written as where and .

Applications of Composition

• Area of an Oil Spill: If the radius of a circular oil slick increases at 2 miles per hour, then and the area . After 6 hours, square miles.

• Car Pricing: If a car's MSRP is , a rebate is , and a dealer discount is . The order of applying these functions affects the final price. For example, and . Applying the rebate first and then the discount costs $320$ more than applying the discount first and then the rebate.

Summary Table: Operations on Functions

Additional info: The examples and applications provided illustrate how combining and composing functions are used in practical scenarios, such as calculating areas and determining pricing strategies.