2. Copyright © 2009 Pearson Education, Inc. 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

3. Copyright © 2009 Pearson Education, Inc. 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.

4. Slide 2.3-4 Copyright © 2009 Pearson Education, Inc. Composition of Functions Definition:

5. Slide 2.3-5 Copyright © 2009 Pearson Education, Inc. Example Given that f(x) = 3x  1 and g(x) = x 2 + x  3, find: a)b) a)

6. Slide 2.3-6 Copyright © 2009 Pearson Education, Inc. Example Given that f(x) = 3x  1 and g(x) = x 2 + x  3, find: a) b) b)

7. Slide 2.3-7 Copyright © 2009 Pearson Education, Inc. Example Given that f(x) = 3x  1 and g(x) = x 2 + x  3, find: a)b) a)

8. Slide 2.3-8 Copyright © 2009 Pearson Education, Inc. Example Given that f(x) = 3x  1 and g(x) = x 2 + x  3, find: a)b) b)

9. Slide 2.3-9 Copyright © 2009 Pearson Education, Inc. 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

10. Slide 2.3-10 Copyright © 2009 Pearson Education, Inc. 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.

11. Slide 2.3-11 Copyright © 2009 Pearson Education, Inc. 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) = x 4 and g(x) = 3x  1.