1 4 Building Functions from Functions Copyright 2011

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1. 4 Building Functions from Functions Copyright © 2011 Pearson, Inc.

1. 4 Building Functions from Functions Copyright © 2011 Pearson, Inc.

What you’ll learn about n n n Combining Functions Algebraically Composition of Functions Relations

What you’ll learn about n n n Combining Functions Algebraically Composition of Functions Relations and Implicitly Defined Functions … and why Most of the functions that you will encounter in calculus and in real life can be created by combining or modifying other functions. Copyright © 2011 Pearson, Inc. 2

Sum, Difference, Product, and Quotient Copyright © 2011 Pearson, Inc. 3

Sum, Difference, Product, and Quotient Copyright © 2011 Pearson, Inc. 3

Example Defining New Functions Algebraically Copyright © 2011 Pearson, Inc. 4

Example Defining New Functions Algebraically Copyright © 2011 Pearson, Inc. 4

Solution Copyright © 2011 Pearson, Inc. 5

Solution Copyright © 2011 Pearson, Inc. 5

Composition of Functions Copyright © 2011 Pearson, Inc. 6

Composition of Functions Copyright © 2011 Pearson, Inc. 6

Composition of Functions Copyright © 2011 Pearson, Inc. 7

Composition of Functions Copyright © 2011 Pearson, Inc. 7

Example Composing Functions Copyright © 2011 Pearson, Inc. 8

Example Composing Functions Copyright © 2011 Pearson, Inc. 8

Solution Copyright © 2011 Pearson, Inc. 9

Solution Copyright © 2011 Pearson, Inc. 9

Example Decomposing Functions Copyright © 2011 Pearson, Inc. 10

Example Decomposing Functions Copyright © 2011 Pearson, Inc. 10

Solution Copyright © 2011 Pearson, Inc. 11

Solution Copyright © 2011 Pearson, Inc. 11

Implicitly Defined Functions The general term for a set of ordered pairs (x, y)

Implicitly Defined Functions The general term for a set of ordered pairs (x, y) is a relation. If the relation happens to relate a single value of y to each value of x, then the relation is also a function. In the case of x 2 + y 2 = 4, it is not a function itself, but we can split it into two equations that do define functions: we say that the relation given by the equation defines the two functions implicitly. Copyright © 2011 Pearson, Inc. 12

Example Using Implicitly Defined Functions Copyright © 2011 Pearson, Inc. 13

Example Using Implicitly Defined Functions Copyright © 2011 Pearson, Inc. 13

Solution Copyright © 2011 Pearson, Inc. 14

Solution Copyright © 2011 Pearson, Inc. 14

Solution (continued) Copyright © 2011 Pearson, Inc. 15

Solution (continued) Copyright © 2011 Pearson, Inc. 15

Quick Review Copyright © 2011 Pearson, Inc. 16

Quick Review Copyright © 2011 Pearson, Inc. 16

Quick Review Solutions Copyright © 2011 Pearson, Inc. 17

Quick Review Solutions Copyright © 2011 Pearson, Inc. 17