Functions and Their Graphs

Introduction

This study guide covers foundational concepts in functions and their graphs, including definitions, properties, notation, domain and range, function operations, and graphical analysis. These topics are essential for success in Precalculus and further mathematical studies.

Relations and Functions

Definitions and Representations

• Relation: A relation is a set of ordered pairs, showing a relationship between two sets (often x and y values).

• Function: A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value).

• Representing Relations: Relations can be represented as tables, mappings (arrows from domain to range), or graphs on the coordinate plane.

• Example: The set { (3, 0), (5, -1) } is a relation.

Domain and Range

Definitions and Notation

• Domain: The set of all possible input values (x-values) for a function.

• Range: The set of all possible output values (y-values) for a function.

• Interval Notation: Uses parentheses and brackets to describe sets of numbers.

  • Parentheses ( ) mean "not included" or "open".

  • Brackets [ ] mean "included" or "closed".

  • Infinity symbols (, ) are always paired with parentheses.

  • Example: means all numbers greater than -2 and up to and including 5.

• Set Notation: Describes the set using inequalities and braces.

  • Example: means all x greater than 0.

  • Symbols and mean "not included"; and mean "included".

Identifying Functions

Vertical Line Test

• The Vertical Line Test is used to determine if a graph represents a function.

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

• Example: A parabola opening upwards passes the vertical line test; a circle does not.

Function Notation

Explicit Functions and Evaluation

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

• Evaluating Functions: Substitute the given value for x into the function.

  • Example: If , then .

Operations with Functions

Sum, Difference, Product, and Quotient

• Sum:

• Difference:

• Product:

• Quotient: , where

• Domain of Combined Functions: The domain is the intersection of the domains of the individual functions, except for the quotient, which excludes values where .

Difference Quotient

Definition and Application

• The difference quotient is used to find the average rate of change of a function and is foundational for calculus.

• Formula:

• Example: For , the difference quotient is .

Graphing Functions

Intercepts and Behavior

• x-intercept: The point(s) where the graph crosses the x-axis ().

• y-intercept: The point where the graph crosses the y-axis ().

• Increasing/Decreasing: A function is increasing where its graph rises as x increases, and decreasing where it falls.

• End Behavior: Describes how the function behaves as or .

Types of Functions: Even and Odd

Symmetry Properties

• Even Function: Satisfies for all x in the domain; graph is symmetric about the y-axis.

• Odd Function: Satisfies for all x in the domain; graph is symmetric about the origin.

• Example: is even; is odd.

Piecewise Functions

Definition and Evaluation

• Piecewise Function: Defined by different expressions for different intervals of the domain.

• Evaluating: Identify which interval the input value belongs to, then use the corresponding expression.

• Graphing: Plot each piece on its interval, paying attention to endpoints and open/closed circles.

• Example:

Tables: Domain and Range Notation Comparison

Purpose: Comparison of Interval and Set Notation

Practice and Application

Sample Problems

• Given a graph, determine the domain and range in set and interval notation.

• Use the vertical line test to identify functions.

• Evaluate functions for given values.

• Find the difference quotient for a given function.

• Classify functions as even, odd, or neither.

• Graph piecewise functions and interpret their behavior.

Additional info: This guide expands on brief worksheet prompts and includes academic context for Precalculus students, ensuring clarity and completeness for exam preparation.