Properties of Functions

Even and Odd Functions

Understanding whether a function is even, odd, or neither is fundamental in precalculus. These properties are determined by the function's symmetry and algebraic behavior.

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

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

Example: The function f(x) = x^2 is even, while f(x) = x^3 is odd.

Identifying Even and Odd Functions from a Graph

To determine if a function is even or odd from its graph, observe the symmetry:

• Y-axis symmetry: Indicates an even function.

• Origin symmetry: Indicates an odd function.

Identifying Even and Odd Functions from an Equation

To check algebraically:

• Substitute -x for x in the function.

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

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

Increasing, Decreasing, and Constant Functions

Definition of Increasing Function

A function f is increasing on an interval I if for any two numbers x_1 and x_2 in I with x_1 < x_2, we have f(x_1) < f(x_2).

• Key Point: The graph rises as you move from left to right.

Definition of Decreasing Function

A function f is decreasing on an interval I if for any two numbers x_1 and x_2 in I with x_1 < x_2, we have f(x_1) > f(x_2).

• Key Point: The graph falls as you move from left to right.

Local and Absolute Extrema

Local Maximum and Minimum

A function f has a local maximum at c if f(c) is greater than or equal to f(x) for all x near c. It has a local minimum at c if f(c) is less than or equal to f(x) for all x near c.

• Local Maximum: Highest point in a neighborhood.

• Local Minimum: Lowest point in a neighborhood.

Example: Finding Local Extrema

Given a graph, identify the x-values where the function attains local maxima and minima, and list their values.

• Local Maximum: At x = 0, f(0) = 3.

• Local Minima: At x = -2 and x = 1.

Intervals of Increase and Decrease

To find where a function is increasing or decreasing, examine the graph:

• Increasing: Where the graph rises.

• Decreasing: Where the graph falls.

Absolute Maximum and Minimum

The absolute maximum of a function is the highest value attained on its domain, and the absolute minimum is the lowest value attained.

• Absolute Maximum: For all x in the domain, f(u) >= f(x).

• Absolute Minimum: For all x in the domain, f(v) <= f(x).

Practice Examples

Example: Intercepts, Domain, and Range

To find intercepts, domain, and range:

• Intercepts: Points where the graph crosses the axes.

• Domain: All possible input values (x-values).

• Range: All possible output values (y-values).

Summary Table: Properties of Functions

Additional info: All explanations have been expanded for clarity and completeness, and images included are directly relevant to the adjacent content.