The Algebra of Functions Section 2.2 The Algebra of Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives 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.

Sums, Differences, Products, and Quotients of Functions If f and g are functions and x is in the domain of each function, then

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)

Example (cont) 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: 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

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.

Example (cont) 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).

Difference Quotient The ratio below is called the difference quotient, or average rate of change.

Example For the function f given by f (x) = 5x  1, find the difference quotient Solution: We first find f (x + h):

Example (cont)

Example For the function f given by f (x) = x2 + 2x  3, find the difference quotient. Solution: We first find f (x + h):

Example(cont)