2.2 More on Functions and Their Graphs

2.2.1 Increasing, Decreasing, and Constant Functions

Understanding how a function behaves as its input changes is fundamental in precalculus. Functions can be classified as increasing, decreasing, or constant over specific intervals.

• Increasing Function: A function f is increasing on an interval I if for any x_1, x_2 \in I with x_1 < x_2, we have f(x_1) < f(x_2). Graphically, the curve rises as you move from left to right.

• Decreasing Function: A function f is decreasing on an interval I if for any x_1, x_2 \in I with x_1 < x_2, we have f(x_1) > f(x_2). The graph falls as you move from left to right.

• Constant Function: A function f is constant on an interval I if for all x_1, x_2 \in I, f(x_1) = f(x_2). The graph is a horizontal line.

Example: Consider the graph below. Identify intervals where the function is increasing, decreasing, or constant.

2.2.2 Relative Maxima and Relative Minima

Relative extrema are points where a function reaches a local highest or lowest value within a certain interval.

• Relative Maximum: A function value f(a) is a relative maximum if there exists an open interval around a such that f(a) > f(x) for all x near a. This is the top of a hill on the graph.

• Relative Minimum: A function value f(b) is a relative minimum if there exists an open interval around b such that f(b) < f(x) for all x near b. This is the bottom of a valley on the graph.

Example: Use the graph below to answer:

• At which x-values does f have a relative maximum?

• What is the relative maximum value?

• Find all relative minima as ordered pairs.

• Determine intervals where the function is increasing, decreasing, or constant.

2.2.3 Even and Odd Functions

Functions can exhibit symmetry, which helps in graphing and understanding their properties.

• Even Function: A function f is even if f(-x) = f(x) for all x in the domain. The graph is symmetric about the y-axis.

• Odd Function: A function f is odd if f(-x) = -f(x) for all x in the domain. The graph is symmetric about the origin.

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

2.2.4 Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Definition: A function f(x) is piecewise if it is defined by multiple rules for different parts of its domain.

Example:

Find f(5) and f(50):

• f(5): Since 0 ≤ 5 ≤ 15, f(5) = 40.

• f(50): Since 50 > 15, f(50) = 40 + 0.60(50 - 15) = 40 + 0.60 \times 35 = 40 + 21 = 61.

Graphing Piecewise Functions: Graph each piece over its interval, using open or closed circles to indicate whether endpoints are included.

2.2.5 Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and the foundation of calculus.

• Definition: For a function f, the difference quotient is:

• Example 1: If f(x) = 3x + 7, find and simplify the difference quotient.

• Example 2: If f(x) = x^2 - 3x + 2, find and simplify the difference quotient.

2.2.6 Graph Analysis Exercise

Study the graph below and answer the following:

• Is the function even, odd, or neither?

• What is the domain and range?

• Find the x-intercept(s) and y-intercept.

• Identify intervals where the function is increasing, decreasing, or constant.

• Find numbers at which the function has relative maxima and minima.

• For which x-values is f(x) = 3?