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

Section 1.3 – Properties of Functions

Learning Objectives

• Determine when a function is odd, even, or neither from a graph or an equation.

• Determine when a function is increasing, decreasing, or constant from a graph.

• Identify local and absolute extrema (maximums or minimums) from a graph.

Odd and Even Functions

Symmetry in Graphs

Graphs of functions may exhibit symmetry with respect to the y-axis, the origin, or neither. This symmetry helps classify functions as even, odd, or neither.

• Even Function: The graph is symmetric with respect to the y-axis.

• Odd Function: The graph is symmetric with respect to the origin.

• Neither: The graph does not exhibit either symmetry.

Note: Some relations (not functions) may be symmetric with respect to the x-axis, but this does not apply to functions.

Definitions

• Even Function: A function f is called even if its graph is symmetric with respect to the y-axis. Algebraically, $f(-x) = f(x)$ for all $x$ in the domain of $f$.

• Odd Function: A function f is called odd if its graph is symmetric with respect to the origin. Algebraically, $f(-x) = -f(x)$ for all $x$ in the domain of $f$.

Identifying Even and Odd Functions Algebraically

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

• Even: $f(-x) = f(x)$

• Odd: $f(-x) = -f(x)$

• Neither: If neither condition is met, the function is neither even nor odd.

Example:

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

• $f(x) = 5x^3 - x$ $f(-x) = 5(-x)^3 - (-x) = -5x^3 + x = -(5x^3 - x) = -f(x)$ Odd function

• $f(x) = x^2 + 6x + 1$ $f(-x) = (-x)^2 + 6(-x) + 1 = x^2 - 6x + 1$ Neither even nor odd

Intervals of Increase, Decrease, and Constancy

Definitions from a Graph

The behavior of a function on an interval can be classified as increasing, decreasing, or constant by examining its graph.

• Increasing: On an open interval $(x_1, x_2)$, $f(x_1) < f(x_2)$ as $x$ increases.

• Decreasing: On an open interval $(x_1, x_2)$, $f(x_1) > f(x_2)$ as $x$ increases.

• Constant: On an open interval $(x_1, x_2)$, $f(x_1) = f(x_2)$ for all $x$ in the interval.

Example: Given a graph of $y = f(x)$, identify intervals where the function is increasing, decreasing, or constant by observing the slope of the graph.

Absolute and Local Extrema

Definitions

Extrema refer to the maximum and minimum values of a function, which can be classified as absolute (global) or local (relative).

Type	Algebraic "Formal" Definition	"Happy" Definition
Absolute Minimum	A function has an absolute minimum at $x = c$ if $f(c) ext{ is } ext{min}ig(f(x)ig)$ for all $x$ in the domain $D$.	The lowest point on the graph over the entire domain.
Absolute Maximum	A function has an absolute maximum at $x = c$ if $f(c) ext{ is } ext{max}ig(f(x)ig)$ for all $x$ in the domain $D$.	The highest point on the graph over the entire domain.
Local Minimum	A function has a local minimum at $x = c$ if $f(c) ext{ is } ext{min}ig(f(x)ig)$ for $x$ near $c$ (in an open interval).	A "valley" point lower than nearby points.
Local Maximum	A function has a local maximum at $x = c$ if $f(c) ext{ is } ext{max}ig(f(x)ig)$ for $x$ near $c$ (in an open interval).	A "peak" point higher than nearby points.

Finding Extrema from a Graph

To find local and absolute extrema, examine the graph for highest and lowest points, both overall and within neighborhoods.

• Local extrema: Look for peaks (local maxima) and valleys (local minima) within intervals.

• Absolute extrema: Identify the single highest and lowest points on the entire graph.

Example: Given a graph, determine:

• Domain of $f$ (using interval notation)

• Local minima and maxima (points where the graph changes direction)

• Absolute minima and maxima (overall lowest and highest points)

Summary Table: Extrema Definitions

Type	Algebraic Definition	Graphical Interpretation
Absolute Minimum	$f(c) ext{ is } ext{min}ig(f(x)ig)$ for all $x$ in $D$	Lowest point on the graph
Absolute Maximum	$f(c) ext{ is } ext{max}ig(f(x)ig)$ for all $x$ in $D$	Highest point on the graph
Local Minimum	$f(c) ext{ is } ext{min}ig(f(x)ig)$ for $x$ near $c$	Valley point
Local Maximum	$f(c) ext{ is } ext{max}ig(f(x)ig)$ for $x$ near $c$	Peak point

Practice and Application

• Given equations, determine if the function is even, odd, or neither using algebraic tests.

• Given graphs, identify intervals of increase, decrease, and constancy.

• Find local and absolute extrema from graphs, and state the domain using interval notation.

Additional info: The notes include graphical examples and practice problems for each concept, reinforcing the definitions and procedures for College Algebra students.