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Question 1: Recall that a function F(t, y) is said to be homogeneous if F(At. Ay) = F(t., y) for any constant λ. Show that the set of all homogeneous functions F(t. y) is closed under addition and multiplication.

Question 1: Recall that a function F(t, y) is said to be homogeneous if F(At. Ay) = F(t., y) for any constant λ. Show that the set of all homogeneous functions F(t. y) is closed under addition and multiplication.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
ChapterA: Appendix
SectionA.2: Geometric Constructions
Problem 9P: A soda can is made from 40 square inches of aluminum. Let x denote the radius of the top of the can,...
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= F(t, y) for any Question 1: Recall that a function F(t, y) is said to be homogeneous if F(At. Ay) : constant 2. Show that the set of all homogeneous functions F(t. y) is closed under addition and multiplication. potitution u-u/t
Transcribed Image Text:= F(t, y) for any Question 1: Recall that a function F(t, y) is said to be homogeneous if F(At. Ay) : constant 2. Show that the set of all homogeneous functions F(t. y) is closed under addition and multiplication. potitution u-u/t
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Definition :  A function Ft,y is said to be homogeneous if  F λt , λy  =F t , y  where λ be a real number.

      Now consider the set  S =  F : F is a homogeneous function 

         Let  F and G ∈ S

         So F and G are two homogeneous functions. So there exists scalars m and n such that

                    F mt , ny  =  F t , y     and    G nt , ny  = G t , y  

 

(1)  To prove S is closed under addition : 

                       F+G λt , λy  = F λt , λy  +G λt , λy  

                                                =   F t , y  +  G t , y 

                                               = F+G  t , y 

                           F+G λt , λy  = F+G  t , y 

                   ⇒  F+G is homogeneous function 

                  ⇒  F+G ∈ S   ∀ F, G ∈ S

                       Therefore S is closed under addition.                     

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