CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry and Transformations 2.5 Variation and Applications Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

2.3 The Composition of Functions Find the composition of two functions and the domain of the composition. Decompose a function as a composition of two functions. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Composition of Functions Definition: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) a) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) b) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) a) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given that f(x) = 3x  1 and g(x) = x2 + x  3, find: a) b) b) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Given , find the domain of Solution: f (x) is not defined for negative radicands. Since the inputs of are the outputs of g, the domain of consists of all the values in the domain of g for which g(x) is nonnegative. The domain is Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Decomposing a Function as a Composition In calculus, one needs to recognize how a function can be expressed as the composition of two functions. This can be thought of as “decomposing” the function. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example If h(x) = (3x  1)4, find f(x) and g(x) such that Solution: The function h(x) raises (3x  1) to the fourth power. Two functions that can be used for the composition are: f(x) = x4 and g(x) = 3x  1. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley