2.3: FUNCTIONS. abs( x)

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1 2.3: FUNCTIONS Definition: Let A and B be sets. A function f is a rule that assigns to each element x A exactly one element y B, written y = f (x). A is called the domain of f and f is said to be defined on A. B is called the codomain of f. The range of f, denoted f (A), is f (A) = {f (x) B x A}. Let S A. The image of S, denoted f (S), is f (S) = {f (x) x S}. Let T B. The preimage of T, denoted f 1 (T), is f 1 (T) = {x A f (x) T}. Example 1: Let A = {1,2,3} and B = {,, }. Let f : A B be defined as f (1)= f (2)= and f (3)=. Determine each of the following. (a) f (A) (b) f (S) if S = {1,2} (c) f 1 (T) if T = { } Example 2: (a) Absolute Value: abs : R R or : R R is a piecewise function defined as x if x 0; abs( x) x. x if x 0. (b) Factorial: Define 0! = 1 and for n Z +, n! = 1 2 n. 96

2 Definition: Functions that convert real numbers to integers. The floor function (a.k.a. the greatest integer function), written x or floor(x) for x R, is the largest integer less than or equal to x. The ceiling function, written x or ceil(x) for x R, is the smallest integer greater than or equal to x. Example 3: How many digits are in the numbers 127, 2 15, and ? Example 4: Solve for x in the equation 5x = 4. Definition: The fractional part function, written frac(x) for x R, is defined as frac(x) = x x. Definition: Let f : A B. The graph of f, denoted graph(f), is the set graph(f ) = { (x,y) A B f (x) = y }. Example 5: Draw the graph of frac(x). Definition: Let A be a set. The function A :A A is defined as A (x) = x for x A is called the identity map on A. 97

3 Definition: Let f : A B. (i) f is one-to-one (or injective) if distinct elements in A have distinct images in B. (ii) f is onto (or surjective) if B = f (A). Example 6: Determine if the following functions are 1-1 or onto or both. (a) Let p: Z Z be defined by (i) p(n) = n+4, (ii) p(n) = n 2, (iii) p(n) = n 3. (b) Let f : Z Z Z be defined by f (m,n) = m+n. Definition: Let f : A B. f is in one-to-one correspondence (or a bijection) if it is both one-to-one and onto. Definition: Given functions f : A B and g: B C, the composition g f : A C is defined as (g f )(x) = g(f (x)). Example 7: For A = {1,2,3}, B= {x,y,z}, and C = {,, }, define f : A B as f = { (1,x),(2,y),(3,x) } and define g : B C as g = { (x, ),(y, ),(z, ) }. What is g f? 98

4 Definition: Let f : A B be a bijection. The inverse of f is the unique function g: B A such that f g : B B is the identity map on B and g f : A A is the identity map on A. (That is, f g B and g f A.) The inverse of f is traditionally written as f 1. (Note that 1/f is different!) A function that has an inverse is said to be invertible. PARTIAL FUNCTIONS Definition: A partial function f from a set A to a set B is an assignment to each element x in a subset of A, called the domain of definition of f, of a unique element y in B. A is the domain of f. B is the codomain of f. f is undefined for elements in A that are not in the domain of definition of f. f is a total function if the domain of definition of f equals A. Example 8: A partial function with domain A = {1,2,3,4,5} and codomain B = {w,x,y,z} is {(1,x), (2,x), (3,y)}. How many partial functions are there? 99

5 SECTION 2.3 FUNCTIONS Definition: Let A and B be sets. A function f is a rule that assigns to each element x A exactly one element y B, written y = f (x). A is called the domain of f and f is said to be defined on A. B is called the codomain of f. The range of f, denoted f (A), is f (A) = {f (x) B x A}. Let S A. The image of S, denoted f (S), is f (S) = {f (x) x S}. Let T B. The preimage of T, denoted f -1 (T), is f -1 (T) = {x A f (x) T}. Example 1: Let A = {1,2,3} and B = {,, }. Let f : A B be defined as f (1) = f (2) = and f (3) =. Determine each of the following. (a) f (A) (b) f (S) if S = {1,2} (c) f -1 (T) if T = { } Example 2: (a) Absolute Value: abs : R R or : R R is a piecewise function defined as x if x 0; abs( x) x. x if x 0. (b) Factorial: Define 0! = 1 and for n Z +, n! = 1 2 n. 100

6 Definition: Functions that convert real numbers to integers. The floor function (a.k.a. the greatest integer function), written x or floor(x) for x R, is the largest integer less than or equal to x. The ceiling function, written x or ceil(x) for x R, is the smallest integer greater than or equal to x. Example 3: How many digits are in the numbers 127, 2 15, and ? Example 4: Solve for x in the equation 5x = 4. Definition: The fractional part function, written frac(x) for x R, is defined as frac(x) = x x Definition: Let f : A B. The graph of f, denoted graph(f), is the set graph(f ) = { (x,y) A B f (x) = y }. 101

7 Example 5: Draw the graph of frac(x). Definition: Let A be a set. The function 1 A :A A is defined as 1 A (x) = x for x A is called the identity map on A. Definition: Let f : A B. (i) f is one-to-one (or injective) if distinct elements in A have distinct images in B. (ii) f is onto (or surjective) if B = f (A). Example 6: Determine if the following functions are 1-1 or onto or both. (a) Let p: Z Z be defined by (i) p(n) = n+4, (ii) p(n) = n 2, (iii) p(n) = n 3. (b) Let f: Z Z Z be defined by f (m,n) = m+n. 102

8 Definition: Let f : A B. f is in one-to-one correspondence (or a bijection) if it is both one-toone and onto. Definition: Given functions f : A B and g: B C defined as (g f )(x) = g(f (x))., the composition g f : A C is Example 7: For A = {1,2,3}, B= {x,y,z}, and C = {,, }, let f : A B be defined as f = { (1,x),(2,y),(3,x) } and let g : B C be defined as g = { (x, ),(y, ),(z, ) }. What is g f? Definition: Let f : A B be a bijection. The inverse of f is the unique function g: B A such that g f = 1 A and f g = 1 B. The inverse of f is traditionally written as f 1. (Note that 1/f is different!) A function that has an inverse is said to be invertible. PARTIAL FUNCTIONS Definition: A partial function f from a set A to a set B is an assignment to each element x in a subset of A, called the domain of definition of f, of a unique element y in B. A is the domain of f. B is the codomain of f. f is undefined for elements in A that are not in the domain of definition of f. f is a total function if the domain of definition of f equals A. 103

9 Example 8: For A = {1,2,3} and B = {x,y,z}, one partial function with domain A and codomain B is {(1,x), (2,y)}. How many partial functions are there? 104