Properties of Functions: Even and Odd Functions, Graph Symmetry, and Algebraic Identification

Functions and Their Graphs

Introduction

This section explores the properties of functions, focusing on the classification of functions as even or odd, their graphical symmetries, and methods for identifying these properties both visually and algebraically. Understanding these concepts is fundamental in College Algebra and aids in analyzing and interpreting function behavior.

Properties of Functions

Objectives

• Identify even and odd functions from a graph

• Identify even and odd functions from an equation

• Use a graph to determine where a function is increasing, decreasing, or constant

• Use a graph to locate local maxima and local minima

• Use a graph to locate the absolute maximum and the absolute minimum

• Use a graphing utility to approximate local maxima and minima and to determine where a function is increasing or decreasing

• Find the average rate of change of a function

Even and Odd Functions

Definition: Even Function

A function f is even if, for every number x in its domain, the number −x is also in the domain and:

$f(-x) = f(x)$

• Even functions are symmetric with respect to the y-axis.

• Common examples: $f(x) = x^2$, $f(x) = ext{cos}(x)$

Definition: Odd Function

A function f is odd if, for every number x in its domain, the number −x is also in the domain and:

$f(-x) = -f(x)$

• Odd functions are symmetric with respect to the origin.

• Common examples: $f(x) = x^3$, $f(x) = ext{sin}(x)$

Theorem: Graphs of Even and Odd Functions

• A function is even if and only if its graph is symmetric with respect to the y-axis.

• A function is odd if and only if its graph is symmetric with respect to the origin.

Example 1: Identifying Even and Odd Functions from a Graph

• Even Function: The graph is symmetric with respect to the y-axis. For example, the parabola $y = x^2$.

• Odd Function: The graph is symmetric with respect to the origin. For example, the cubic function $y = x^3$.

Example 2: Identifying Even and Odd Functions Algebraically

Determine whether each function is even, odd, or neither. Also, determine the symmetry of the graph.

Function	Even/Odd/Neither	Symmetry
$f(x) = 2x^2 + 3$	Even	y-axis
$g(x) = x^3 + 3$	Neither	None
$h(x) = x^3 + 4x$	Odd	Origin
$F(x) = 1 + |x|$	Even	y-axis

• To test algebraically, substitute $-x$ for $x$ and compare $f(-x)$ to $f(x)$ and $-f(x)$.

• If $f(-x) = f(x)$, the function is even.

• If $f(-x) = -f(x)$, the function is odd.

• If neither, the function is neither even nor odd.

Example: Detailed Algebraic Checks

• $f(x) = 2x^2 + 3$: $f(-x) = 2(-x)^2 + 3 = 2x^2 + 3 = f(x)$ (Even)

• $g(x) = x^3 + 3$: $g(-x) = (-x)^3 + 3 = -x^3 + 3 eq g(x)$ and $g(-x) eq -g(x)$ (Neither)

• $h(x) = x^3 + 4x$: $h(-x) = (-x)^3 + 4(-x) = -x^3 - 4x = -h(x)$ (Odd)

• $F(x) = 1 + |x|$: $F(-x) = 1 + |-x| = 1 + |x| = F(x)$ (Even)

Summary Table: Even and Odd Function Properties

Property	Even Function	Odd Function
Algebraic Test	$f(-x) = f(x)$	$f(-x) = -f(x)$
Graph Symmetry	y-axis	Origin
Example	$x^2$, $|x|$	$x^3$, $ext{sin}(x)$