Functions and Their Graphs

Even and Odd Functions

Understanding whether a function is even, odd, or neither is fundamental in precalculus. These properties are determined both algebraically and graphically, and they relate to the symmetry of the function's graph.

• 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 .

• Graphical Symmetry:

  • Even functions are symmetric with respect to the y-axis.

  • Odd functions are symmetric with respect to the origin.

• Example: is even because .

• Example: is odd because .

Increasing, Decreasing, and Constant Functions

The behavior of a function on an interval can be classified as increasing, decreasing, or constant. These properties are essential for analyzing and graphing functions.

• Increasing Function: f is increasing on interval I if, for any in I, .

• Decreasing Function: f is decreasing on interval I if, for any in I, .

• Constant Function: f is constant on interval I if, for all x in I, is equal.

Local Maxima and Minima

Local maxima and minima are points where a function reaches a highest or lowest value within a neighborhood. These are important for understanding the shape and turning points of a graph.

• 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.

Absolute Maximum and Minimum

The absolute maximum and minimum are the highest and lowest values a function attains on a given interval. These are also called global extrema.

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

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

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval , then it must attain both an absolute maximum and an absolute minimum on that interval.

• Continuity: For precalculus, a function is considered continuous if its graph has no gaps or holes and can be drawn without lifting the pencil.

Using Graphing Utilities to Approximate Extrema and Intervals of Increase/Decrease

Graphing calculators and software can be used to find approximate values for local maxima and minima, and to determine where a function is increasing or decreasing.

• Example: For on , the graphing utility finds a local maximum at with , and a local minimum at with .

• Intervals: The function is increasing on and decreasing on and .

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. It is analogous to the slope of a line connecting two points on the graph.

• Definition: For a function and interval , the average rate of change is:

• Example: For :

  • From $0:

  • From $1:

  • From $0:

Slope of a Secant Line

The slope of the secant line between two points on a function's graph is equal to the average rate of change over that interval. The equation of the secant line can be found using the point–slope form.

• Point–Slope Form: , where is the slope and is a point on the line.

• Example: For , between and , the slope is $1y = x - 1$.