The Algebra of Functions Section 2.2 The Algebra of Functions

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 + 1 and g(x) = find each of the following. a) (f + g)(x) b) (f + g)(6) c) (f + g)(− 4) Solution: a) This cannot be simplified.

Example Given that f(x) = x + 1 and g(x) = find each of the following. a) (f + g)(x) b) (f + g)(6) c) (f + g)(− 4) Solution: b)

Example Given that f(x) = x + 1 and g(x) = find each of the following. a) (f + g)(x) b) (f + g)(6) c) (f + g)(− 4) Solution: c) We must first determine whether – 4 is in the domain of each function. We note that – 4 is not in the domain of g, thus, (f + g)(− 4) does not exist.

Domains of f + g, f – g, fg, AND f/g If f and g are functions, then the domain of the functions f + g, f – g, and fg is the intersection of the domain of f and the domain of g. The domain of f/g is also the intersection of the domains of f and g with the exclusion of any x-values for which g(x) = 0.

Example Given that f(x) = x2 − 4 and g(x) = x + 2, find each of the following. a) The domain of f + g, f  g, fg, and f/g 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 −2 , since g(−2) = 0.

Example continued b) (f + g)(x) = f(x) + g(x) = (x2 − 4) + (x + 2) = x2 + x + −2 c) d)

Example continued e) f)

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

Example For the function f given by f (x) = 2x  3, find the difference quotient Solution:

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

Example(cont)