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Macroeconomics Exam Review 97

# Macroeconomics Exam Review 97 - Solutions for Foundations...

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Chapter 3: Linear Functions 3.1 Let x 1 , x 2 𝑋 and 𝛼 1 , 𝛼 2 ∈ ℜ . Homogeneity implies that 𝑓 ( 𝛼 1 x 1 ) = 𝛼 1 𝑓 ( 𝑥 1 ) 𝑓 ( 𝛼 2 x 2 ) = 𝛼 2 𝑓 ( 𝑥 2 ) and additivity implies 𝑓 ( 𝛼 1 x 1 + 𝛼 2 x 2 ) = 𝛼 1 𝑓 ( x 1 ) + 𝛼 2 𝑓 ( x 2 ) Conversely, assume 𝑓 ( 𝛼 1 x 1 + 𝛼 2 x 2 ) = 𝛼 1 𝑓 ( x 1 ) + 𝛼 2 𝑓 ( x 2 ) for all x 1 , x 2 𝑋 and 𝛼 1 , 𝛼 2 ∈ ℜ . Letting 𝛼 1 = 𝛼 2 = 1 implies 𝑓 ( x 1 + x 2 ) = 𝑓 ( x 1 ) + 𝑓 ( x 2 ) while setting x 2 = 0 implies 𝑓 ( 𝛼 1 x 1 ) = 𝛼 1 𝑓 ( x 1 ) 3.2 Assume 𝑓 1 , 𝑓 2 𝐿 ( 𝑋, 𝑌 ). Define the mapping 𝑓 1 + 𝑓 2 : 𝑋 𝑌 by ( 𝑓 1 + 𝑓 2 )( x ) = 𝑓 1 ( x ) + 𝑓 2 ( x ) We have to confirm that 𝑓 1 + 𝑓 2 is linear, that is ( 𝑓 1 + 𝑓 2 )( x 1 + x 2 ) = 𝑓 1 ( x 1 + x 2 ) + 𝑓 2 ( x 1 + x 2 ) = 𝑓 1 ( x 1 ) + 𝑓 1 ( x 2 ) + 𝑓 2 ( x 1 ) + 𝑓 2 ( x 2 ) = 𝑓 1 ( x 1 ) + 𝑓 2 ( x 1 ) + 𝑓 1 ( x 1 ) + 𝑓 2 ( x
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