Functions and Graphs

Combinations of Functions; Composite Functions

This section explores how to combine functions using algebraic operations and how to form composite functions. Understanding domains is essential for both combining and composing functions, as certain values must be excluded to avoid undefined expressions.

Finding a Function’s Domain

The domain of a function is the set of all real numbers for which the function is defined. When a function does not model specific data or verbal conditions, its domain is determined by excluding values that cause division by zero or result in an even root of a negative number.

• Division by zero: Any value of x that makes the denominator zero must be excluded.

• Even roots of negative numbers: Values that result in a negative number under an even root (such as a square root) must be excluded.

Example: If a function g(x) has a denominator of (x^2 - 49), then x = 7 and x = -7 must be excluded from the domain because they make the denominator zero.

The Algebra of Functions: Sum, Difference, Product, and Quotient

Given two functions f and g, you can combine them using the following operations. The domain of each combined function is the set of real numbers common to the domains of f and g, with additional restrictions for the quotient.

• Sum:

• Difference:

• Product:

• Quotient: , where

Example: If and both have unrestricted domains, then their sum, difference, and product also have unrestricted domains. For the quotient, exclude values where .

The Composition of Functions

The composition of two functions f and g, denoted , is defined as $. The domain of the composite function consists of all x such that:

• x is in the domain of g

• g(x) is in the domain of f

Example: If and , then . The domain is because must be non-negative.

Excluding Values from the Domain of Composite Functions

When forming composite functions, certain values must be excluded:

• If x is not in the domain of g, it must not be in the domain of .

• If g(x) is not in the domain of f, x must also be excluded from the domain of .

Example: If is undefined for , then is not in the domain of . If is undefined for , then is also excluded.

Writing Functions as Compositions

Some functions can be expressed as compositions of two or more functions. For example, can be written as where and .

Summary Table: Operations on Functions