Functions and Graphs

Combinations of Functions; Composite Functions

This section explores how to find the domain of a function, combine functions using algebraic operations, form composite functions, and determine the domains of these combinations. Understanding these concepts is essential for analyzing and constructing more complex mathematical models.

Finding a Function’s Domain

• Domain of a Function: The domain is the largest set of real numbers for which the function produces real values.

• Restrictions: Exclude values that cause division by zero or result in an even root (such as a square root) of a negative number.

• Example: If a function has a denominator, set the denominator not equal to zero and solve for excluded values.

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

• Sum:

• Difference:

• Product:

• Quotient: , where

• Domain: The domain of each operation is the set of all real numbers common to the domains of and , with additional restrictions for the quotient (where ).

• Example: If and are both defined for all real numbers, then so are their sum, difference, and product. For the quotient, exclude values where .

The Composition of Functions

• Definition: The composition of with is written as .

• Domain of Composite Function: The domain of is the set of all such that $x$ is in the domain of and is in the domain of .

• Example: If is defined for all real numbers, but is only defined for , then is only defined for those where .

Excluding Values from the Domain of Composite Functions

• Step 1: Exclude any not in the domain of .

• Step 2: Exclude any for which is not in the domain of .

• Summary: The domain of is all such that $x$ is in the domain of and is in the domain of .

Writing a Function as a Composition

• Decomposition: Any function can sometimes be written as a composition of two simpler functions and , such that .

• Example: If , then and , so .