mathCharFunct-07

# mathCharFunct-07 - ARE211, Fall 2007 CHARFUNCT: TUE, NOV 6,...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ARE211, Fall 2007 CHARFUNCT: TUE, NOV 6, 2007 PRINTED: DECEMBER 14, 2007 (LEC# 20) Contents 5. Characteristics of Functions. 1 5.1. Surjective, Injective and Bijective functions 1 5.2. Homotheticity 3 5.3. Homogeneity and Euler’s theorem 4 5.4. Monotonic functions 5 5.5. Concave and quasi-concave functions; Definiteness, Hessians and Bordered Hessians. 5 5. Characteristics of Functions. 5.1. Surjective, Injective and Bijective functions Definition: A function f : X → A is onto or surjective if every point in the range is reached by f starting from some point in the domain, i.e., if ∀ a ∈ A , ∃ x ∈ X such that f ( x ) = a . Example: consider f : R → R defined by f ( x ) = 5 for all x . This function is not surjective because there are points in the range (i.e., R ) that aren’t reachable by f (i.e., any point in R except 5 is unreachable). Definition: A function f : X → A is 1-1 or injective if for every point in the range, there is at most one point in the domain that gets mapped by f to that point. More precisely, f is injective if for all x,x prime such that x negationslash = x prime , f ( x ) negationslash = f ( x prime ) . 1 2 CHARFUNCT: TUE, NOV 6, 2007 PRINTED: DECEMBER 14, 2007 (LEC# 20) Example: consider f : R → R defined by f ( x ) = 5 for all x . This function is not injective because there is a point in the range (i.e., 5) that gets mapped to by f , starting from a large number of points in the domain. Definition: A function f : X → A is bijective if it is surjective and injective. Definition: A function f : X → A is invertible if there exists a function f- 1 : A → X such that for all a ∈ A , f ( f- 1 ( a )) = a and for all x ∈ X , f- 1 ( f ( x )) = x . (We’ll show below that you need both of these conditions to capture the notion of invertibility.) If such a function exists, it is called the inverse of f. Theorem: A function f : X → A is invertible if and only if it is bijective. • You need f to be surjective (onto): otherwise there would be points in the range that aren’t mapped to by f from any point in the domain: E.g., take f : [0 , 1] → R defined by f ( x ) = 5: the image of [0 , 1] under f is { 5 } Now consider any point a negationslash∈ { 5 } . Since f doesn’t map anything to a , there cannot exist a function f- 1 such that f ( f- 1 ( a ) ) = a . • You need f to be injective (1-1): otherwise the inverse couldn’t be a function. E.g., take f : [0 , 1] → [01 / 4] defined by f ( x ) = x- x 2 and take a = 3 / 16; this point is reached from both 1 / 4 and 3 / 4, i.e., f (1 / 4) = f (3 / 4) = 3 / 16. Now apply the definition to both 1 / 4 and 3 / 4: it is required that for all x ∈ X , f- 1 ( f ( x )) = x . In particular, it is required that f- 1 ( f (1 / 4)) = f- 1 (3 / 16) = 1 / 4. But it is also required that f- 1 ( f (3 / 4)) = f- 1 (3 / 16) = 3 / 4. Thus f is required to map the same point to two different places, and hence can’t be a function....
View Full Document

## This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at Berkeley.

### Page1 / 12

mathCharFunct-07 - ARE211, Fall 2007 CHARFUNCT: TUE, NOV 6,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online