Section 2.3: Properties of Functions

Even and Odd Functions

Understanding whether a function is even, odd, or neither is fundamental in analyzing its symmetry and behavior. These properties can be determined both graphically and algebraically.

• Even Function: A function f is even if, for every number x in its domain, –x is also in the domain and .

• Odd Function: A function f is odd if, for every number x in its domain, –x is also in the domain and .

Theorem: The graph of an even function is symmetric with respect to the y-axis, while the graph of an odd function is symmetric with respect to the origin.

Examples:

• Example 1 (Algebraic): Determine if the following functions are even, odd, or neither:

  • (even)

  • (neither)

  • (odd)

  • (even)

Example 2 (Graphical): To determine evenness or oddness from a graph, check for y-axis or origin symmetry, respectively.

Increasing, Decreasing, and Constant Functions

The behavior of a function over an interval can be classified as increasing, decreasing, or constant. These properties are essential for understanding the overall trend of a function.

• Increasing: A function f is increasing on an interval I if, for any in I, .

• Decreasing: A function f is decreasing on an interval I if, for any in I, .

• Constant: A function f is constant on an interval I if, for all in I, is equal.

Example: Given a graph, identify intervals where the function is increasing, decreasing, or constant by observing the slope and direction of the curve.

Local Maximum and Minimum (Local Extrema)

Local extrema are points where a function reaches a highest or lowest value within a certain interval. These are important for analyzing the behavior of functions in calculus and optimization problems.

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

Example: Use the graph of a function to identify the x-values where local maxima and minima occur, and list their corresponding function values.

Absolute Maximum and Minimum (Global Extrema)

The absolute maximum and minimum are the highest and lowest values a function attains on a given interval. These are crucial for understanding the range and limits of a function.

• Absolute Maximum: If there is a number u in I such that for all x in I, then is the absolute maximum.

• Absolute Minimum: If there is a number v in I such that for all x in I, then is the absolute minimum.

Extreme Value Theorem: If a function is continuous on a closed interval [a, b], it must have both an absolute maximum and an absolute minimum on that interval.

Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input. This concept is foundational for calculus and real-world applications such as velocity and growth rates.

• Definition: For a function , the average rate of change from to is: , where .

Example: For :

• From 0 to 1:

• From 1 to 2:

• From 0 to 2:

Slope of a Secant Line: The average rate of change between and equals the slope of the secant line passing through and on the graph of .

Summary Table: Properties of Functions