Properties of Functions: Even/Odd Functions, Intervals, Extrema, and Average Rate of Change

Section 1.3 Properties of Functions

Objectives Overview

• Identify even and odd functions from a graph and from an equation.

• Determine where a function is increasing, decreasing, or constant using a graph.

• Locate local and absolute maxima and minima from a graph.

• Use a graphing utility to approximate extrema and intervals of increase/decrease.

• Find the average rate of change of a function.

Even and Odd Functions

Definition of Even Function

An even function is a function f such that for every number x in its domain, −x is also in the domain and:

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

Key Property: The graph of an even function is symmetric with respect to the y-axis.

Definition of Odd Function

An odd function is a function f such that for every number x in its domain, −x is also in the domain and:

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

Key Property: The graph of an odd function is symmetric with respect to the origin.

Theorem: Symmetry and Function Type

• 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.

Examples: Determining Even and Odd Functions from Graphs

• If a graph is symmetric about the y-axis, the function is even.

• If a graph is symmetric about the origin, the function is odd.

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

Determining Even and Odd Functions Algebraically

• To test if f is even, compute f(−x) and compare to f(x).

• To test if f is odd, compute f(−x) and compare to −f(x).

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

Intervals of Increase, Decrease, and Constancy

Definitions

• Increasing: A function f is increasing on an interval I if, for any choice of x_1 < x_2 in I, f(x_1) < f(x_2).

• Decreasing: f is decreasing on I if, for any x_1 < x_2 in I, f(x_1) > f(x_2).

• Constant: f is constant on I if, for all x in I, f(x) is the same value.

Example: Identifying Intervals from a Graph

Given a graph, determine where the function is increasing, decreasing, or constant by observing the slope of the curve:

• Increasing: The graph rises as you move left to right.

• Decreasing: The graph falls as you move left to right.

• Constant: The graph is flat (horizontal line).

Local Maxima and Minima

Definitions

• Local Maximum: f has a local maximum at c if there is an open interval containing c such that f(c) ≥ f(x) for all x in that interval.

• Local Minimum: f has a local minimum at c if there is an open interval containing c such that f(c) ≤ f(x) for all x in that interval.

Example: Finding Local Extrema from a Graph

To find local maxima and minima, look for peaks (maximum) and valleys (minimum) in the graph.

Absolute Maximum and Minimum

Definitions

• Absolute Maximum: f has an absolute maximum at u if f(u) ≥ f(x) for all x in the interval.

• Absolute Minimum: f has an absolute minimum at v if f(v) ≤ f(x) for all x in the interval.

Example: Identifying Absolute Extrema from a Graph

Check endpoints and any local extrema to determine the absolute maximum and minimum values.

Theorem: Extreme Value Theorem

If f is a continuous function on a closed interval [a, b], then f has both an absolute maximum and an absolute minimum on [a, b].

Using a Graphing Utility for Extrema and Intervals

Procedure

• Graph the function using a graphing calculator or software.

• Use built-in commands to approximate local maxima and minima.

• Identify intervals of increase and decrease by observing the graph.

Average Rate of Change of a Function

Definition

The average rate of change of a function f from a to b is:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Example: Calculating Average Rate of Change

For f(x) = x^2 + 1:

• From 0 to 2: $\frac{f(2) - f(0)}{2 - 0} = \frac{5 - 1}{2} = 2$

• From 1 to 2: $\frac{f(2) - f(1)}{2 - 1} = \frac{5 - 2}{1} = 3$

Theorem: Slope of the Secant Line

The average rate of change of a function from a to b equals the slope of the secant line through the points (a, f(a)) and (b, f(b)) on its graph.

Example: Equation of a Secant Line

Given g(x) and two points (1, g(1)) and (3, g(3)):

• Find the average rate of change (slope): $m = \frac{g(3) - g(1)}{3 - 1}$

• Use the point–slope form: $y - g(1) = m(x - 1)$

• Convert to slope–intercept form if needed.

Additional info: These concepts are foundational for understanding calculus, where the average rate of change leads to the concept of the derivative, and local/absolute extrema are critical for optimization problems.