Functions and Their Graphs

Definition of Relations and Functions

A relation is a correspondence between two sets, typically called the domain and range. In a relation, each element from the domain is associated with at least one element from the range.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

• If a relation assigns exactly one output to each input, it is called a function.

• Notation: If y depends on x, we write .

Definition: Let X and Y be two nonempty sets. A function from X into Y is a relation that associates each element of X with exactly one element of Y.

Identifying Functions

To determine whether a relation is a function, check if each input has only one output.

• Vertical Line Test: If any vertical line crosses the graph more than once, the relation is not a function.

• Example: is not a function because input 3 has two outputs.

• Example: is a function because each input has one output.

Function Notation and Variables

Functions are often described as machines that take an input from the domain and produce an output in the range.

• Independent variable: The input variable (usually x).

• Dependent variable: The output variable (usually y or f(x)), which depends on the input.

• Function notation: , where x is the independent variable.

Implicit and Explicit Forms of Functions

• Explicit form: The function is solved for the dependent variable (e.g., ).

• Implicit form: The function is not solved for the dependent variable (e.g., ).

Difference Quotient

The difference quotient of a function f at x is given by:

• Used in calculus to find derivatives.

• In precalculus, focus on simplifying the expression.

Finding the Domain of a Function

To find the domain of a function defined by an equation:

• Start with all real numbers.

• If the equation has a denominator, exclude values that make the denominator zero.

• If the equation has a radical with an even index, exclude values that make the radicand negative.

Examples:

• : Exclude x-values that make denominator zero (, ). Domain:

• : Exclude . Domain:

• : Require . Domain:

Function Operations

Functions can be combined using addition, subtraction, multiplication, and division.

• Difference: is defined by

• Sum: is defined by

• Product: is defined by

• Quotient: is defined by ,

• The domain of each operation is the intersection of the domains of and , with additional restrictions for division.

Evaluating Functions

To evaluate a function, substitute the input value into the function expression.

• Example: If , then

• Example:

Application: Domain in Word Problems

When finding the domain in application problems, consider the context. For example, if a function models the area of a rectangle in terms of width, only positive values make sense.

• Example: for

Summary Table: Function Operations and Domains

Check Your Understanding

• Determine if a relation is a function by checking if each input has only one output.

• Find the domain of a function by considering restrictions from denominators and radicals.

• Apply function operations and determine the domain of the resulting function.

Additional info: These notes are based on a high school/college-level precalculus curriculum and include both definitions and worked examples for foundational concepts in functions and their graphs.