1. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 6.1 Composite Functions

2. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

4. Composition of Functions The composition combines two functions by applying one function to a number and then applying the other function to the result. Domain of g Range of g Domain of f Range of f X f ₀ g fg g(x)g(x) f (g(x)) = (f ₀ g)(x) Copyright © 2009 Carol A. Marinas, Ph.D.

5. Composition of Functions The function h(x) = (3x + 1) 2 is the composition combines two functions. The first function is 3x + 1 and the second function is to square the result. So g(x) = 3x + 1 and f(x) = x 2. X h(x) = (f ₀ g)(x) = (3x + 1) 2 f g g(x) = 3x + 1 (f ₀ g)(x) = f (g(x)) = f (3x + 1) = (3x + 1) 2 Copyright © 2009 Carol A. Marinas, Ph.D.

6. Composition of Functions With the function h(x) = (3x + 1) 2, find h(5) using the two- step composition method. 5 h(5) = (f ₀ g)(5) = (16) 2 = 256 f g g(x) = 35 + 1 = 16 (f ₀ g)(x) = f (g(x)) = f (16) = (16) 2 h(5) = 256 Copyright © 2009 Carol A. Marinas, Ph.D.

7. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Like an Assembly Line

8. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

9. Find: (a) f ⁰ g (b) g ⁰ f Finding a Composite Function

10. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find: (a) f ⁰ g (b) g ⁰ f Finding a Composite Function

11. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find: (a) f ⁰ g (b) f ⁰ f Finding a Composite Function

12. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find: (a) f ⁰ g (b) f ⁰ f Finding a Composite Function

13. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.