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.

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VocabularyStep Function

Greatest Integer Function

Even and Odd Functions

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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.

a. Graph the model over the appropriate domain using your graphing calculator.

b. Use this graph to determine which years the population was increasing. During which years was the population decreasing?

c. Approximate the maximum population between 1995 and 2001.

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Greatest Integer FunctionDefined as the greatest integer less than or

equal to x.

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More Greatest IntegerThe greatest integer function is an example

of a step function.

Find the following values:

5.1 c.

10

1 b.

1 a.

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Graphing a Piecewise Function

Sketch the graph of f (x) by hand.

1,4

1,32)(

xx

xxxf

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Even and Odd FunctionsEven 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). 8

Graphs of Function Symmetry

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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).

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Example 6Determine algebraically and graphically

whether each function is even, odd, or

neither.

a. g(x) = x3 – x

b. h(x) = x2 + 1

c. f (x) = x3 – 1

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

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