SECTION 2.1 Functions 71 When we use functions in applications, the domain may be restricted by physical or geometric considerations. For example, the domain of the function f defined by ( ) = f x x2 is the set of all real numbers. However, if f represents the area of a square whose sides are of length x, the domain of f is restricted to the positive real numbers, since the length of a side can never be 0 or negative. Finding the Domain of a Function Used in an Application Express the area of a circle as a function of its radius. Find the domain. Solution EXAMPLE 10 See Figure 16. The formula for the area A of a circle of radius r is π = A r .2 Using r to represent the independent variable and A to represent the dependent variable, the function expressing this relationship is π ( ) = = A A r r2 In this application, the domain is { } > r r 0 . (Do you see why?) Figure 16 Circle of radius r A r Now Work PROBLEM 105 6 Form the Sum, Difference, Product, and Quotient of Two Functions Functions, like numbers, can be added, subtracted, multiplied, and divided. For example, if ( ) = + f x x 9 2 and ( ) = + g x x3 5, then ( ) ( ) ( ) ( ) + = + + + = + + f x g x x x x x 9 3 5 3 14 2 2 The new function = + + y x x3 14 2 is called the sum function + f g. Similarly, ( ) ( ) ( ) ( ) ⋅ = + + = + + + f x g x x x x x x 9 3 5 3 5 27 45 2 3 2 The new function = + + + y x x x 3 5 27 45 3 2 is called the product function ⋅ f g. The general definitions are given next. DEFINITION Sum Function Given functions f and g, the sum function is defined by ( )( ) ( ) ( ) + = + f g x f x g x DEFINITION Difference Function Given functions f and g, the difference function is defined by ( )( ) ( ) ( ) − = − f g x f x g x The domain of + f g consists of all real numbers x that are in the domains of both f and g. That is, domain of + = ∩ f g f g domain of domain of . The domain of − f g consists of all real numbers x that are in the domains of both f and g. That is, domain of − = ∩ f g f g domain of domain of . DEFINITION Product Function Given functions f and g, the product function is defined by ( )( ) ( ) ( ) ⋅ = ⋅ f g x f x g x The domain of ⋅ f g consists of all real numbers x that are in the domains of both f and g. That is, domain of ⋅ = ∩ f g f g domain of domain of . Recall The symbol ∩ stands for intersection. It means the set of elements that are common to both sets. j
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