1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 1 Chapter 5 Logarithmic Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 2 5.1 Composite Functions

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 3 Composite Function Definition If f and g are functions, x is in the domain of g, and g(x) is in the domain of f, then we can form the composite function f ◦ g: (f ◦ g)(x) = f(g(x)) We say f ◦ g is the composition of f and g.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 4 Example: Using Tables to Evaluate a Composite Function All of the input-output pairs of functions g and f are shown in the tables below. 1. Find (f ◦ g)(0). 2. Use a table to describe five input-output pairs of f ◦ g.

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 5 Solution 1. 2. We repeat the process used in Problem 1 for the inputs 1, 2, 3, and 4 and how the work in the table.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 6 Example: Evaluating a Composite Function Let f(x) = 2x – 5 and g(x) = 3x + 6. 1. Find (f ◦ g)(4).2. Find (g ◦ f)(4).

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 7 Solution 1. 2. Let f(x) = 2x – 5 and g(x) = 3x + 6.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 8 Composite Functions In general, the outputs (f ◦ g)(a) and (g ◦ f)(a) may or may not be equal.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 9 Example: Finding Equations of Composition Functions Let f(x) = 2 x and g(x) = x + 4. 1. Find an equation of f ◦ g. 2. Find an equation of g ◦ f.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 10 Solution 1. Since g(x) = x + 4, we can substitute x + 4 for g(x) in the second step:

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 11 Solution 1. We verify our work by creating a graphing calculator table for y = f(g(x)) and y = 2 x + 4.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 12 Solution 2. Since f(x) = 2 x, we can substitute 2 x for f(x) in the second step.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 13 Example: Evaluating and Finding an Equation of a Composite Function Let f(x) = –3x + 8 and g(x) = 4x – 5. 1. Find (f ◦ g)(2). 2. Find an equation of f ◦ g. 3. Use the equation of f ◦ g to find (f ◦ g)(2).

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 14 Solution 1. Let f(x) = –3x + 8 and g(x) = 4x – 5.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 15 Solution 2. Since g(x) = 4x – 5, we can substitute 4x – 5 for g(x) in the second step. Let f(x) = –3x + 8 and g(x) = 4x – 5.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 16 Solution 3. So, (f ◦ g)(2) is equal to –1, which is the same result we found in Problem 1. Let f(x) = –3x + 8 and g(x) = 4x – 5.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 17 Example: Expressing a Function as a Composition of Two Functions If h(x) = (7x + 3) 5, find equations of the functions f and g such that h(x) = (f ◦ g)(x).

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 18 Solution To form f(g(x)), we substitute g(x) for x in f(x). Similarly, to form (7x + 3) 5, we can substitute 7x + 3 for x in x 5. This suggests that g(x) = 7x + 3 and f(x) = x 5. We check by performing the composition: (f ◦ g)(x) = f(g(x)) = f(7x + 3) = (7x + 3) 5

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 19 Solution There are other possible answers. For example, g(x) = 7x and f(x) = (x + 3) 5 also work: (f ◦ g)(x) = f(g(x)) = f(7x) = (7x + 3) 5

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 20 Example: Using Graphs to Evaluate a Composite Function Refer to the graph to find (f ◦ g)(5).

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 21 Solution The blue arrows show that g(5) = –3. So, we can substitute –3 for g(5) in the second step:

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 22 Example: Using a Composite Function to Model a Situation Let f(Q) be the number of cups in Q quarts, and let g(x) be the number of ounces in x cups. 1. Find an equation of f. 2. Find an equation of g. 3. Find an equation of g ◦ f. 4. Find (g ◦ f)(5). What does it mean in this situation?

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 23 Solution 1. Since there are 4 cups in one quart, f(Q) = 4Q. 2. Since there are 8 ounces in one cup, g(x) = 8x. 3. (g ◦ f)(Q) = (g(f(Q)) = g(4Q) = 8(4Q) = 32Q

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 24 Solution 4. (g ◦ f)(5) = 32(5) = 160 The function f converts units of quarts to units of cups, and the function g converts units of cups to units of ounces, so g ◦ f converts units of quarts to units of ounces. So, (g ◦ f)(5) = 160 means there are 160 ounces in 5 quarts.