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

2.2 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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) (f + g)(x)= f (x)+g(x) =x+2+2x+5 =3x+7 b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15 (f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or (f + g)(5) = 3(5) + 7 = 22 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Given that f(x) = x2 + 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Another Example continued b) (f  g)(x) = f(x)  g(x) = (x2 + 2)  (x  3) = x2  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). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Difference Quotient The ratio below is called the difference quotient, or average rate of change. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the function f given by f (x) = 5x  1, find the difference quotient Solution: We first find f (x + h): Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example continued Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example For the function f given by f (x) = x2 + 2x  3, find the difference quotient. Solution: We first find f (x + h): Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example continued Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley