Properties of Functions: Even and Odd Functions, Increasing/Decreasing Intervals, and Extrema

Section 3.3: Properties of Functions

Objectives

• 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 maxima and minima, as well as absolute maximum and minimum, from a graph

• Use a graphing utility to approximate local 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, the number -x is also in the domain and:

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

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

• Common examples: $f(x) = x^2$, $f(x) = |x|$

Definition of Odd Function

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

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

• The graph of an odd function is symmetric with respect to the origin.

• Common examples: $f(x) = x^3$, $f(x) = an(x)$

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

• Neither Even nor Odd: If the graph is neither symmetric with respect to the y-axis nor the origin, the function is neither even nor odd.

• Even Function: If the graph is symmetric with respect to the y-axis, the function is even.

• Odd Function: If the graph is symmetric with respect to the origin, the function is odd.

Example 2: Identifying Even and Odd Functions Algebraically

• To determine if a function is even, odd, or neither, substitute -x for x and compare $f(-x)$ to $f(x)$ and $-f(x)$.

Function	Even/Odd/Neither	Reasoning
$f(x) = 2x^2 + 3$	Even	$f(-x) = 2(-x)^2 + 3 = 2x^2 + 3 = f(x)$
$g(x) = x^3 + 3$	Neither	$g(-x) = (-x)^3 + 3 = -x^3 + 3 \neq g(x)$ and $\neq -g(x)$
$h(x) = x^3 + 4x$	Odd	$h(-x) = (-x)^3 + 4(-x) = -x^3 - 4x = -h(x)$
$F(x) = 1 + |x|$	Even	$F(-x) = 1 + |-x| = 1 + |x| = F(x)$

Increasing, Decreasing, and Constant Functions

Definitions

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

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

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

Example: Determining Intervals of Increase, Decrease, and Constancy

• Use the graph to identify intervals where the function rises (increasing), falls (decreasing), or remains flat (constant).

• Express intervals using interval notation, e.g., $[-2, 0]$.

Local and Absolute Extrema

Local Maximum and Minimum

• Local Maximum: $f$ has a local maximum at $c$ if $f(c) \geq f(x)$ for all $x$ near $c$.

• Local Minimum: $f$ has a local minimum at $c$ if $f(c) \leq f(x)$ for all $x$ near $c$.

• The values $f(c)$ are called local maximum or minimum values.

Absolute Maximum and Minimum

• Absolute Maximum: $f$ has an absolute maximum at $u$ if $f(u) \geq f(x)$ for all $x$ in the domain.

• Absolute Minimum: $f$ has an absolute minimum at $v$ if $f(v) \leq f(x)$ for all $x$ in the domain.

Extreme Value Theorem

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

• Additional info: Continuity means the graph has no breaks, jumps, or holes.

Example: Finding Extrema from a Graph

• Identify the highest and lowest points on the graph within the domain.

• Check for local maxima/minima at turning points and endpoints (if the interval is closed).

Using Graphing Utilities

Approximating Maxima, Minima, and Intervals of Increase/Decrease

• Graphing calculators or software can estimate the coordinates of local maxima and minima.

• Use features like 'MAXIMUM' and 'MINIMUM' to find these points numerically.

• Analyze the graph to determine intervals where the function is increasing or decreasing.

Average Rate of Change

Definition

The average rate of change of a function $f$ from $a$ to $b$ (where $a \neq b$) is:

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

• This measures how much $f(x)$ changes per unit change in $x$ over the interval $[a, b]$.

Example: Calculating Average Rate of Change

• For $f(x) = x^2 + 1$ from $0$to $1$:

• $f(1) = 2$, $f(0) = 1$

• Average rate: $\frac{2 - 1}{1 - 0} = 1$

Slope of a Secant Line

Theorem

• The average rate of change of $f$ from $a$ to $b$ equals the slope of the secant line through $(a, f(a))$ and $(b, f(b))$.

$m_{secant} = \frac{f(b) - f(a)}{b - a}$

Example: Equation of a Secant Line

• Given $g(x) = x^2 - 3x + 2$, find the average rate of change from $x=1$ to $x=3$:

• $g(1) = 0$, $g(3) = 2$

• Slope: $m = \frac{2 - 0}{3 - 1} = 1$

• Equation using point-slope form: $y - 0 = 1(x - 1) \implies y = x - 1$