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Functions and Relations Technique: Math Basic
- 1. 07/04/25 1 … … andthe following mathematical and the following mathematical appetizer is about… appetizer is about… Functions Functions
- 2. 07/04/25 2 Functions Functions A A function functionf from a set A to a set B is an f from a set A to a set B is an assignment assignment of exactly one element of B to each of exactly one element of B to each element of A. element of A. We write We write f(a) = b f(a) = b if b is the unique element of B assigned by the if b is the unique element of B assigned by the function f to the element a of A. function f to the element a of A. If f is a function from A to B, we write If f is a function from A to B, we write f: A f: A B B (note: Here, “ (note: Here, “ “ has nothing to do with if… then) “ has nothing to do with if… then)
- 3. 07/04/25 3 Functions Functions If f:A Iff:A B, we say that A is the B, we say that A is the domain domain of f and B of f and B is the is the codomain codomain of f. of f. If f(a) = b, we say that b is the If f(a) = b, we say that b is the image image of a and a is of a and a is the the pre-image pre-image of b. of b. The The range range of f:A of f:A B is the set of all images of B is the set of all images of elements of A. elements of A. We say that f:A We say that f:A B B maps maps A to B. A to B.
- 4. 07/04/25 4 Functions Functions Let ustake a look at the function f:P Let us take a look at the function f:P C with C with P = {Linda, Max, Kathy, Peter} P = {Linda, Max, Kathy, Peter} C = {Boston, New York, Hong Kong, Moscow} C = {Boston, New York, Hong Kong, Moscow} f(Linda) = Moscow f(Linda) = Moscow f(Max) = Boston f(Max) = Boston f(Kathy) = Hong Kong f(Kathy) = Hong Kong f(Peter) = New York f(Peter) = New York Here, the range of f is C. Here, the range of f is C.
- 5. 07/04/25 5 Functions Functions Let usre-specify f as follows: Let us re-specify f as follows: f(Linda) = Moscow f(Linda) = Moscow f(Max) = Boston f(Max) = Boston f(Kathy) = Hong Kong f(Kathy) = Hong Kong f(Peter) = Boston f(Peter) = Boston Is f still a function? Is f still a function? yes yes {Moscow, Boston, Hong Kong} {Moscow, Boston, Hong Kong} What is its range? What is its range?
- 6. 07/04/25 6 Functions Functions Other waysto represent f: Other ways to represent f: Boston Boston Peter Peter Hong Hong Kong Kong Kathy Kathy Boston Boston Max Max Moscow Moscow Linda Linda f(x) f(x) x x Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow
- 7. 07/04/25 7 Functions Functions If thedomain of our function f is large, it is If the domain of our function f is large, it is convenient to specify f with a convenient to specify f with a formula formula, e.g.: , e.g.: f: f:R R R R f(x) = 2x f(x) = 2x This leads to: This leads to: f(1) = 2 f(1) = 2 f(3) = 6 f(3) = 6 f(-3) = -6 f(-3) = -6 … …
- 8. 07/04/25 8 Functions Functions Let f Letf1 1 and f and f2 2 be functions from A to be functions from A to R R. . Then the Then the sum sum and the and the product product of f of f1 1 and f and f2 2 are also are also functions from A to functions from A to R R defined by: defined by: (f (f1 1 + f + f2 2)(x) = f )(x) = f1 1(x) + f (x) + f2 2(x) (x) (f (f1 1f f2 2)(x) = f )(x) = f1 1(x) f (x) f2 2(x) (x) Example: Example: f f1 1(x) = 3x, f (x) = 3x, f2 2(x) = x + 5 (x) = x + 5 (f (f1 1 + f + f2 2)(x) = f )(x) = f1 1(x) + f (x) + f2 2(x) = 3x + x + 5 = 4x + 5 (x) = 3x + x + 5 = 4x + 5 (f (f1 1f f2 2)(x) = f )(x) = f1 1(x) f (x) f2 2(x) = 3x (x + 5) = 3x (x) = 3x (x + 5) = 3x2 2 + 15x + 15x
- 9. 07/04/25 9 Functions Functions We alreadyknow that the We already know that the range range of a function of a function f:A f:A B is the set of all images of elements a B is the set of all images of elements a A. A. If we only regard a If we only regard a subset subset S S A, the set of all A, the set of all images of elements s images of elements s S is called the S is called the image image of S. of S. We denote the image of S by f(S): We denote the image of S by f(S): f(S) = {f(s) | s f(S) = {f(s) | s S} S}
- 10. 07/04/25 10 Functions Functions Let uslook at the following well-known function: Let us look at the following well-known function: f(Linda) = Moscow f(Linda) = Moscow f(Max) = Boston f(Max) = Boston f(Kathy) = Hong Kong f(Kathy) = Hong Kong f(Peter) = Boston f(Peter) = Boston What is the image of S = {Linda, Max} ? What is the image of S = {Linda, Max} ? f(S) = {Moscow, Boston} f(S) = {Moscow, Boston} What is the image of S = {Max, Peter} ? What is the image of S = {Max, Peter} ? f(S) = {Boston} f(S) = {Boston}
- 11. 07/04/25 11 Properties ofFunctions Properties of Functions A function f:A A function f:A B is said to be B is said to be one-to-one one-to-one (or (or injective injective), if and only if ), if and only if x, y x, y A (f(x) = f(y) A (f(x) = f(y) x = y) x = y) In other words: In other words: f is one-to-one if and only if it f is one-to-one if and only if it does not map two distinct elements of A onto the does not map two distinct elements of A onto the same element of B. same element of B.
- 12. 07/04/25 12 Properties ofFunctions Properties of Functions And again… And again… f(Linda) = Moscow f(Linda) = Moscow f(Max) = Boston f(Max) = Boston f(Kathy) = Hong Kong f(Kathy) = Hong Kong f(Peter) = Boston f(Peter) = Boston Is f one-to-one? Is f one-to-one? No, Max and Peter are No, Max and Peter are mapped onto the same mapped onto the same element of the image. element of the image. g(Linda) = Moscow g(Linda) = Moscow g(Max) = Boston g(Max) = Boston g(Kathy) = Hong Kong g(Kathy) = Hong Kong g(Peter) = New York g(Peter) = New York Is g one-to-one? Is g one-to-one? Yes, each element is Yes, each element is assigned a unique assigned a unique element of the image. element of the image.
- 13. 07/04/25 13 Properties ofFunctions Properties of Functions How can we prove that a function f is one-to-one? How can we prove that a function f is one-to-one? Whenever you want to prove something, first take Whenever you want to prove something, first take a look at the relevant definition(s): a look at the relevant definition(s): x, y x, y A (f(x) = f(y) A (f(x) = f(y) x = y) x = y) Example: Example: f: f:R R R R f(x) = x f(x) = x2 2 Disproof by counterexample: f(3) = f(-3), but 3 f(3) = f(-3), but 3 -3, so f is not one-to-one. -3, so f is not one-to-one.
- 14. 07/04/25 14 Properties ofFunctions Properties of Functions … … and yet another example: and yet another example: f: f:R R R R f(x) = 3x f(x) = 3x One-to-one: One-to-one: x, y x, y A (f(x) = f(y) A (f(x) = f(y) x = y) x = y) To show: To show: f(x) f(x) f(y) whenever x f(y) whenever x y y x x y y 3x 3x 3y 3y f(x) f(x) f(y), f(y), so if x so if x y, then f(x) y, then f(x) f(y), that is, f is one-to-one. f(y), that is, f is one-to-one.
- 15. 07/04/25 15 Properties ofFunctions Properties of Functions A function f:A A function f:A B with A,B B with A,B R is called R is called strictly strictly increasing increasing, if , if x,y x,y A (x < y A (x < y f(x) < f(y)), f(x) < f(y)), and and strictly decreasing strictly decreasing, if , if x,y x,y A (x < y A (x < y f(x) > f(y)). f(x) > f(y)). Obviously, a function that is either strictly Obviously, a function that is either strictly increasing or strictly decreasing is increasing or strictly decreasing is one-to-one one-to-one. .
- 16. 07/04/25 16 Properties ofFunctions Properties of Functions A function f:A A function f:A B is called B is called onto onto, or , or surjective surjective, if , if and only if for every element b and only if for every element b B there is an B there is an element a element a A with f(a) = b. A with f(a) = b. In other words, f is onto if and only if its In other words, f is onto if and only if its range range is is its its entire codomain entire codomain. . A function f: A A function f: A B is a B is a one-to-one correspondence one-to-one correspondence, , or a or a bijection bijection, if and only if it is both one-to-one , if and only if it is both one-to-one and onto. and onto. Obviously, if f is a bijection and A and B are finite Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|. sets, then |A| = |B|.
- 17. 07/04/25 17 Properties ofFunctions Properties of Functions Examples: Examples: In the following examples, we use the arrow In the following examples, we use the arrow representation to illustrate functions f:A representation to illustrate functions f:A B. B. In each example, the complete sets A and B are In each example, the complete sets A and B are shown. shown.
- 18. 07/04/25 18 Properties ofFunctions Properties of Functions Is f injective? Is f injective? No. No. Is f surjective? Is f surjective? No. No. Is f bijective? Is f bijective? No. No. Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow
- 19. 07/04/25 19 Properties ofFunctions Properties of Functions Is f injective? Is f injective? No. No. Is f surjective? Is f surjective? Yes. Yes. Is f bijective? Is f bijective? No. No. Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow Paul Paul
- 20. 07/04/25 20 Properties ofFunctions Properties of Functions Is f injective? Is f injective? Yes. Yes. Is f surjective? Is f surjective? No. No. Is f bijective? Is f bijective? No. No. Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow L Lü übeck beck
- 21. 07/04/25 21 Properties ofFunctions Properties of Functions Is f injective? Is f injective? No! f is not even No! f is not even a function! a function! Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow L Lü übeck beck
- 22. 07/04/25 22 Properties ofFunctions Properties of Functions Is f injective? Is f injective? Yes. Yes. Is f surjective? Is f surjective? Yes. Yes. Is f bijective? Is f bijective? Yes. Yes. Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York New York Hong Kong Hong Kong Moscow Moscow L Lü übeck beck Helena Helena
- 23. 07/04/25 23 Inversion Inversion An interestingproperty of bijections is that An interesting property of bijections is that they have an they have an inverse function inverse function. . The The inverse function inverse function of the bijection f:A of the bijection f:A B is B is the function f the function f-1 -1 :B :B A with A with f f-1 -1 (b) = a whenever f(a) = b. (b) = a whenever f(a) = b.
- 24. 07/04/25 24 Inversion Inversion Example: Example: f(Linda) =Moscow f(Linda) = Moscow f(Max) = Boston f(Max) = Boston f(Kathy) = Hong Kong f(Kathy) = Hong Kong f(Peter) = L f(Peter) = Lü übeck beck f(Helena) = New York f(Helena) = New York Clearly, f is bijective. Clearly, f is bijective. The inverse function f The inverse function f- - 1 1 is given by: is given by: f f-1 -1 (Moscow) = Linda (Moscow) = Linda f f-1 -1 (Boston) = Max (Boston) = Max f f-1 -1 (Hong Kong) = Kathy (Hong Kong) = Kathy f f-1 -1 (L (Lü übeck) = Peter beck) = Peter f f-1 -1 (New York) = Helena (New York) = Helena Inversion is only Inversion is only possible for bijections possible for bijections (= invertible functions) (= invertible functions)
- 25. 07/04/25 25 Inversion Inversion Linda Linda Max Max Kathy Kathy Peter Peter Boston Boston New York NewYork Hong Kong Hong Kong Moscow Moscow L Lü übeck beck Helena Helena f f f f-1 -1 f f-1 -1 :C :C P is no P is no function, because function, because it is not defined it is not defined for all elements of for all elements of C and assigns two C and assigns two images to the pre- images to the pre- image New York. image New York.
- 26. 07/04/25 26 Composition Composition The The composition compositionof two functions g:A of two functions g:A B and B and f:B f:B C, denoted by f C, denoted by f g, is defined by g, is defined by (f (f g)(a) = f(g(a)) g)(a) = f(g(a)) This means that This means that • first first, function g is applied to element a , function g is applied to element a A, A, mapping it onto an element of B, mapping it onto an element of B, • then then, function f is applied to this element of , function f is applied to this element of B, mapping it onto an element of C. B, mapping it onto an element of C. • Therefore Therefore, the composite function maps , the composite function maps from A to C. from A to C.
- 27. 07/04/25 27 Composition Composition Example: Example: f(x) =7x – 4, g(x) = 3x, f(x) = 7x – 4, g(x) = 3x, f: f:R R R R, g: , g:R R R R (f (f g)(5) = f(g(5)) = f(15) = 105 – 4 = 101 g)(5) = f(g(5)) = f(15) = 105 – 4 = 101 (f (f g)(x) = f(g(x)) = f(3x) = 21x - 4 g)(x) = f(g(x)) = f(3x) = 21x - 4
- 28. 07/04/25 28 Composition Composition Composition ofa function and its inverse: Composition of a function and its inverse: (f (f-1 -1 f)(x) = f f)(x) = f-1 -1 (f(x)) = x (f(x)) = x The composition of a function and its inverse The composition of a function and its inverse is the is the identity function identity function i(x) = x. i(x) = x.
- 29. 07/04/25 29 Graphs Graphs The The graph graphof a function of a function f:A f:A B is the set of B is the set of ordered pairs {(a, b) | a ordered pairs {(a, b) | a A and f(a) = b}. A and f(a) = b}. The graph is a subset of A The graph is a subset of A B that can be B that can be used to visualize f in a two-dimensional used to visualize f in a two-dimensional coordinate system. coordinate system.
- 30. 07/04/25 30 Floor andCeiling Functions Floor and Ceiling Functions The The floor floor and and ceiling ceiling functions map the real functions map the real numbers onto the integers ( numbers onto the integers (R R Z Z). ). The The floor floor function assigns to r function assigns to r R R the largest the largest z z Z Z with z with z r, denoted by r, denoted by r r . . Examples: Examples: 2.3 2.3 = 2, = 2, 2 2 = 2, = 2, 0.5 0.5 = 0, = 0, -3.5 -3.5 = -4 = -4 The The ceiling ceiling function assigns to r function assigns to r R R the smallest the smallest z z Z Z with z with z r, denoted by r, denoted by r r . . Examples: Examples: 2.3 2.3 = 3, = 3, 2 2 = 2, = 2, 0.5 0.5 = 1, = 1, -3.5 -3.5 = -3 = -3