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: The set of all real numbers for which a function is defined and produces real values.

• To find the domain, exclude any real numbers 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. For functions with even roots, set the radicand greater than or equal to zero.

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

Given two functions f and g, you can combine them as follows:

• Sum:

• Difference:

• Product:

• Quotient: , where

The domain of each combination is the set of all real numbers common to the domains of f and g, with the additional restriction for the quotient that .

Example: Combining Functions

• Let and .

• Sum:

• Product:

• Domain: Both and are defined for all real numbers, so the domain is .

The Composition of Functions

The composition of two functions f and g is written as . This means you first apply g to x, then apply f to the result.

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

Example: Forming Composite Functions

• Given and , find .

• Solution:

• Domain:

Excluding Values from the Domain of Composite Functions

• If is not in the domain of , it must be excluded from the domain of .

• If is not in the domain of , then must also be excluded from the domain of .

Example: Forming a Composite Function and Finding Its Domain

• Given and , find and its domain.

• Solution:

• Domain: Exclude such that or

• Domain:

Writing a Function as a Composition

• Some functions can be expressed as the composition of two or more simpler functions.

• Example: can be written as where and .