Chapter 1 Functions and Their Graphs Pre-Calculus Chapter 1 Functions and Their Graphs

1.3.2 Even and Odd Functions Objectives: Identify and graph step functions and other piecewise-defined functions. Identify even and odd functions.

Vocabulary Step Function Greatest Integer Function Even and Odd Functions

Warm Up 1.3.2 During a seven-year period, the population P (in thousands) of North Dakota increased and then decreased according to the model P = –0.76t2 + 9.9t + 618, 5 ≤ t ≤ 11, where t represents the year, with t = 5 corresponding to 1995. Graph the model over the appropriate domain using your graphing calculator. Use this graph to determine which years the population was increasing. During which years was the population decreasing? Approximate the maximum population between 1995 and 2001.

Greatest Integer Function Defined as the greatest integer less than or equal to x.

More Greatest Integer The greatest integer function is an example of a step function. Find the following values:

Graphing a Piecewise Function Sketch the graph of f (x) by hand.

Even and Odd Functions Even Function Symmetric with respect to the y-axis. For every (x, y) on the graph, there is also (–x, y). Odd Function Symmetric with respect to the origin. For every (x, y) on the graph, there is also (x, –y).

Graphs of Function Symmetry

How Do We Know If It’s Even, Odd, or Neither? Even Function Graphical: Symmetric about y-axis (mirror image). Algebraic: For each x in domain of f, f (–x) = f (x). Odd Function Graphical: Image is the same when rotated 180°. Algebraic: For each x in domain of f, f (–x) = –f (x).

Example 6 Determine algebraically and graphically whether each function is even, odd, or neither. g(x) = x3 – x h(x) = x2 + 1 f (x) = x3 – 1

Homework 1.3.2 Worksheet 1.3.2 # 41, 45, 47, 53 – 71 odd, 79, 80