Definition: Let f be a real-valued function of a real variable and let M be any real number. The function Mf, called the multiple of f by M or M times f, is the real-valued function with the same domain as f that is defined by the rule Mf(x) = M(f(x)) for each x in the domain of f. Assume that f is a real-valued function of a real variable, and prove the following statement. If f is increasing on a set S and if M is any negative real number, then Mf is decreasing on S. function on a set S of real numbers, and suppose M is any negative Proof: Suppose f is any increasing number. [We will show that Mf is decreasing on S.] Let x₁ and x₂ be any real numbers in S such that x₁ < x₂. Since f is increasing on S, f(x₁)< f(x₂). Multiplying both sides by M gives that M(xi) negative X M(x₁) ---Select--- -f(x₁) M(x₁) M(f(x)) M(x₂) X because M is --Select--- ---Select--- -f(x₁) -f(x=) M(x1) M(x₂) M(f(x:)) M(f(x=)) V By definition of Mf, (MF)(x₁) = ---Select--- and (MF) (x₂) = Thus, by substitution, (Mf)(x₁) * (MF) (x₂). Since x₁ and x₂ were arbitrarily chosen real numbers in S such that x₁ < x₂, we can conclude that Mf is decreasing on S. V real
Definition: Let f be a real-valued function of a real variable and let M be any real number. The function Mf, called the multiple of f by M or M times f, is the real-valued function with the same domain as f that is defined by the rule Mf(x) = M(f(x)) for each x in the domain of f. Assume that f is a real-valued function of a real variable, and prove the following statement. If f is increasing on a set S and if M is any negative real number, then Mf is decreasing on S. function on a set S of real numbers, and suppose M is any negative Proof: Suppose f is any increasing number. [We will show that Mf is decreasing on S.] Let x₁ and x₂ be any real numbers in S such that x₁ < x₂. Since f is increasing on S, f(x₁)< f(x₂). Multiplying both sides by M gives that M(xi) negative X M(x₁) ---Select--- -f(x₁) M(x₁) M(f(x)) M(x₂) X because M is --Select--- ---Select--- -f(x₁) -f(x=) M(x1) M(x₂) M(f(x:)) M(f(x=)) V By definition of Mf, (MF)(x₁) = ---Select--- and (MF) (x₂) = Thus, by substitution, (Mf)(x₁) * (MF) (x₂). Since x₁ and x₂ were arbitrarily chosen real numbers in S such that x₁ < x₂, we can conclude that Mf is decreasing on S. V real
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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