HW4 - i) Is f ◦ g increasing? ii) Is f · g increasing?...

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MATH 186 – WINTER 2009 HOMEWORK SET 4 Be kind to your grader, please staple your work! What you need to remember: - Injective, surjective, and bijective functions - Inverse of a function - Continuity, monotonicity, and injectivity ( see also Exercise 8.8 in Spivak ) - Relationship between matrices and linear transformations What you shouldn’t forget: - Some discontinuous functions are integrable! - Fundamental Theorem of Calculus Ex 1. Let f,g : R R be two injective functions. i) Show that f g is injective. ii) Express ( f g ) - 1 in terms of f - 1 , and g - 1 . iii) Find g - 1 in terms of f - 1 when g is given by g ( x ) = 1 + f ( x ). Ex 2. Let f,g : R R be two increasing functions.

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Unformatted text preview: i) Is f ◦ g increasing? ii) Is f · g increasing? iii) Suppose furthermore that f ( x ) ≥ 0 for all x ∈ R . Describe the graph of f-1 . 2 MATH 186 – WINTER 2009 HOMEWORK SET 4 Ex 3. Let T : R 2 → R 2 be a linear map, and let A ∈ M 2 × 2 ( R ) be its associated matrix, i.e., T ( ~v ) = A~v , where A = a b c d . Let B = { ~w 1 , ~w 2 } be a basis for R 2 . Under which conditions on A is it true that { T ( ~w 1 ) ,T ( ~w 2 ) } is a basis for R 2 ? ( Hint: review Chapter 1.8 in Kaplan & Lewis . ... ) From Spivak: Chapter 12: Ex # 9, 11. Chapter 14: Ex # 4 (ii). Please recycle!...
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This note was uploaded on 11/22/2010 for the course MATH 186 taught by Professor Staff during the Winter '08 term at University of Michigan.

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HW4 - i) Is f ◦ g increasing? ii) Is f · g increasing?...

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