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) b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true. (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, and fg b) (f  g)(x) c) The domain of (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. 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) = Must exclude 3, since (f/g)(x) does not exist at x = 3. Domain: {x | x ≠ 3 } 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 f (x) = x2 + 2x  3 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley