Functions and Their Graphs

Section 1.1: Functions

This section introduces the foundational concepts of relations and functions, including their definitions, notation, and examples. Understanding these concepts is essential for analyzing mathematical relationships and their graphical representations.

• Relation: A relation is a correspondence between two sets: a set X (domain) and a set Y (range). Each element from the domain corresponds to at least one element from the range.

• Domain: The set of all possible input values (independent variable) for a relation or function.

• Range: The set of all possible output values (dependent variable) for a relation or function.

• Function: A function from X into Y is a relation that associates each element of X with exactly one element of Y. In function notation, this is written as .

• Function Notation: The notation denotes the value of the function f at input x.

• Independent Variable: The variable representing the input of the function (commonly x).

• Dependent Variable: The variable representing the output of the function (commonly y), which depends on the value of the independent variable.

Example of a Relation: The pairing of students and their grades. Example of a Function: The relationship between hours spent on physical therapy and improvement in knee flexion.

Determining Functions from Relations

To determine if a relation is a function, check if each input is associated with exactly one output.

Explanation: In the diagram above, 'Bob' is paired with both 'Beth' and 'Diane', so this relation is not a function because one input (Bob) corresponds to more than one output.

Domain and Range in Set Notation

• Example: For the set {(1,3), (2,3), (3,3), (4,3)}, the domain is {1,2,3,4} and the range is {3}. This is a function because each input has only one output.

• Example: For the set {(-4,4), (-3,3), (-2,2), (-1,1), (-4,0)}, the domain is {-4,-3,-2,-1} and the range is {4,3,2,1,0}. This is not a function because -4 is paired with both 4 and 0.

Determining Functions from Equations

• To determine if an equation defines y as a function of x, solve for y and check if each x yields only one y value.

• Examples:

  • is a function (domain: all real numbers).

  • is a function (domain: ).

  • is not a function (a circle fails the vertical line test).

  • is not a function (each yields two values).

  • is a function (domain: ).

Evaluating Functions and Function Operations

• Given , evaluate:

Difference Quotient

• The difference quotient for a function is:

• For : Difference quotient:

• For : Difference quotient:

Operations with Functions

• Given and :

• Sum:

• Difference:

• Product:

• Quotient: , domain excludes where

Application Example

• Gross wages as a function of hours worked at : Domain: (cannot work negative hours)

Section 1.2: The Graph of a Function

This section focuses on graphical representations of functions, including the vertical line test, intercepts, and symmetry.

• Vertical Line Test: A set of points in the -plane is the graph of a function if and only if every vertical line intersects the graph at most once.

• Zero: The zeros of a function are the -values where (i.e., the $x$-intercepts).

Using the Vertical Line Test

• If any vertical line crosses a graph more than once, the graph does not represent a function.

Explanation: The graph above fails the vertical line test, so it does not represent a function.

Explanation: The graph above passes the vertical line test and represents a function with domain .

Explanation: The graph above passes the vertical line test and represents a function with domain .

Explanation: The graph above passes the vertical line test and represents a function with domain .

Analyzing Graphs of Functions

• To analyze a graph, identify:

  • Domain and range

  • - and -intercepts

  • Symmetry (about -axis, -axis, or origin)

  • Where the function is positive or negative

  • Zeros of the function

Explanation: The graph above can be used to answer questions about function values, intercepts, and intervals where the function is positive or negative.

Section 1.3: Properties of Functions

This section explores the classification of functions based on their symmetry and behavior, as well as concepts such as maxima, minima, and average rate of change.

• Even Function: is even if for all in the domain. The graph is symmetric about the -axis.

• Odd Function: is odd if for all in the domain. The graph is symmetric about the origin.

• Increasing Function: As increases, increases.

• Decreasing Function: As increases, decreases.

• Constant Function: As increases, remains unchanged.

• Local Maximum: The highest point in a small neighborhood of the graph.

• Local Minimum: The lowest point in a small neighborhood of the graph.

• Absolute Maximum: The highest value of the function over its entire domain.

• Absolute Minimum: The lowest value of the function over its entire domain.

• Average Rate of Change: The change in over the change in on an interval :

Explanation: The graph above illustrates local maxima, local minima, and absolute extrema.

Even, Odd, or Neither

• To determine if a function is even, odd, or neither, substitute for and compare to and .

Explanation: The graph above is symmetric about the origin, indicating an odd function.

Explanation: The graph above is neither symmetric about the -axis nor the origin, so it is neither even nor odd.

Explanation: The graph above is symmetric about the -axis, indicating an even function.

Average Rate of Change and Secant Lines

• For , the average rate of change from to is:

• The equation of the secant line through and is: , where is the average rate of change.

Summary Table: Properties of Functions

Additional info: Some explanations and examples were expanded for clarity and completeness, including step-by-step calculations for function operations and the difference quotient.