Functions and Graphs: More on Functions and Their Graphs (2.2)

Functions and Graphs: More on Functions and Their Graphs

This chapter explores additional concepts related to functions and their graphs, focusing on how a function's behavior changes, its symmetry properties, and special types of functions like piecewise and difference quotients.

Objectives

• Identify intervals where a function increases, decreases, or is constant.

• Use graphs to locate relative maxima or minima.

• Test for symmetry with respect to the y-axis, x-axis, or the origin.

• Identify even or odd functions and recognize their corresponding symmetries.

• Understand and use piecewise functions.

• Find and simplify a function's difference quotient.

Increasing, Decreasing, and Constant Functions

These definitions describe the behavior of a function's graph over an open interval $I$:

1. Increasing Function: A function $f$ is increasing on an open interval, $I$, if for any two numbers $x<em>1$ and $x</em>2$ in the interval, whenever x1 < x2, then f(x1) < f(x2).

2. Decreasing Function: A function $f$ is decreasing on an open interval, $I$, if for any two numbers $x<em>1$ and $x</em>2$ in the interval, whenever x1 < x2, then f(x1) > f(x2).

3. Constant Function: A function $f$ is constant on an open interval, $I$, if for any two numbers $x<em>1$ and $x</em>2$ in the interval, $f(x<em>1) = f(x</em>2)$.

Example 1: Identifying Intervals of Increase, Decrease, or Constant Behavior

Consider a graph that visually demonstrates these behaviors:

• The function is increasing on the interval $(-\infty, -1)$.

• The function is decreasing on the interval $(-1, 1)$.

• The function is increasing on the interval $(1, \infty)$.

Definitions of Relative Maximum and Relative Minimum

These terms describe the