1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Objective: Students will be able to determine combinations and compositions of two given functions. Students will be able to decompose a given function as the composition of simpler functions. Essential Question: What is the difference between a combination of two functions and a composition of two functions? Home Learning: p #1, 2, 5, 7, 12, 13, 19, 24, 27, 36 & 37

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1.4 Building Functions from Functions Demana, Waits, Foley, Kennedy

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. What you’ll learn about 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.

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Sum, Difference, Product, and Quotient

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Defining New Functions Algebraically

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Composition of Functions

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Composition of Functions

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Composing Functions

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Decomposing Functions

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 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.

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Using Implicitly Defined Functions

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution (continued)