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. 2.2 The Algebra of Functions  Find the sum, the difference, the product, and the quotient of two functions, and determine the domains of the resulting functions.  Find the difference quotient for a function.

4. Slide 2.2-4 Copyright © 2009 Pearson Education, Inc. Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then

5. Slide 2.2-5 Copyright © 2009 Pearson Education, Inc. Example Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following. a) (f + g)(x)b) (f + g)(5) Solution: a) b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or (f + g)(5) = 3(5) + 7 = 22

6. Slide 2.2-6 Copyright © 2009 Pearson Education, Inc. Another Example Given that f(x) = x 2 + 2 and g(x) = x  3, find each of the following. a) The domain of f + g, f  g, fg, and f/g b) (f  g)(x) c) (f/g)(x) Solution: a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g, f  g, and fg are the set of numbers in the intersection of the domains—that is, the set of numbers in both domains, or all real numbers. For f/g, we must exclude 3, since g(3) = 0.

7. Slide 2.2-7 Copyright © 2009 Pearson Education, Inc. Another Example continued b) (f  g)(x) = f(x)  g(x) = (x 2 + 2)  (x  3) = x 2  x + 5 c) (f/g)(x) = Remember to add the stipulation that x  3, since 3 is not in the domain of (f/g)(x).

8. Slide 2.2-8 Copyright © 2009 Pearson Education, Inc. Difference Quotient The ratio below is called the difference quotient, or average rate of change.

9. Slide 2.2-9 Copyright © 2009 Pearson Education, Inc. Example For the function f given by f (x) = 5x  1, find the difference quotient Solution: We first find f (x + h):

10. Slide 2.2-10 Copyright © 2009 Pearson Education, Inc. Example continued

11. Slide 2.2-11 Copyright © 2009 Pearson Education, Inc. Another Example For the function f given by f (x) = x 2 + 2x  3, find the difference quotient. Solution: We first find f (x + h):

12. Slide 2.2-12 Copyright © 2009 Pearson Education, Inc. Example continued