1. Functions Reading: Chapter 6 (94 – 107) from the text book 1

2. What is a function? A function is an input/output rule defined on some set. Example : the rule that takes as input a student in this class, and produces as output that student’s age. Intuitively: a function from A to B is a way to transform each element of A into an element of B 2

3. What is a function? Two conditions must be satisfied: 1- Each input must produce some output. 2- A single input will always produce not more than one output. 3

4. Relations and functions A function can be thought of as a special kind of relation – we get this relation by simply taking the set of all pairs: (input, output) for every possible input. Formally: a function from A to B is a relation on A × B such that for all x ∈ A there exists exactly one element y of B such that (x, y) is in the relation 4

5. Functions as lists of pairs Let A = {1, 2, 3} and B = {a, b, c, d, e}. Consider the relation g ⊆ A × B : g = {(1, d), (2, c), (3, c)} This is a function – each element of A is paired with a single element of B (both 2 and 3 are paired with c but that’s allowed). 5

6. Functions as lists of pairs By contrast h = {(1, a), (3, c), (1, d)} is not a function, because 1 is paired with two things, and 2 isn’t paired with anything. Exercise : Which of the following relations on {a, b, c} × {1, 2, 3} are functions: R 1 = {(a, 1), (a, 2), (b, 3), (c, 2)} R 2 = {(a, 1), (b, 2), (c, 1)} R 3 = {(a, 1), (c, 2)} 6

7. Function notation When we have a function such as g defined previously, we often write it in the following way: g : A ⟶B (g from A to B) g(1) = dg(2) = cg(3) = d Read “g of 1 is equal to d” and so on. This notation is only used for functions and not for more general relations. 7

8. Arrow diagram Arrow diagrams are used to depict functions for which input sets are finite. Example : Let A={1, 2, 3} and B={1, 2, 3, 4}. Let f : A B, f(1)=3, f(2)=2, f(3)=2. Draw an arrow diagram for this function. A B 1 2 3 1 2 3 4 8

9. Functions and formulas It is actually rare to specify a function as a list of pairs. More usually, we give a rule or formula that makes it possible to work out the value (output) of a function on any given element of its domain(input). For instance, we can define a function f ⊆R×R by the formula: f(x) = x 2 + 3 9

10. Functions and formulas To work out the value of the function we just substitute in the formula: f(4) = 4 2 + 3 = 16 + 3 = 19 or in another way of writing the same thing (4, 19)  f. We could also write: f = {(x, x 2 + 3)  x  R} 10

11. Domain, codomain and range The domain of a function is the set of its allowed inputs. The co-domain of a function is the set in which we guarantee its outputs lie. The range of a function is the actual set of its outputs. 11

12. Domain, codomain and range To see the difference between the last two concepts consider f : R → R defined by f(x) = x 2 Its co-domain is R but its range is the set of non-negative real numbers. 12

13. Terminology Let f be a function from A to B. Let x ∈ A The image of x under f, denoted by f(x), is the only element of B such that (x, f(x)) ∈ f Intuitively, f transforms x into f(x) The domain of f is A the codomain of f is B The range of f is {f(x) : x ∈ A} 13

14. Identity and constant functions If the output of a function is always the same, no matter what the input is, then the function is called a constant function. Example f : R→R defined by f(x) = 42 for all x. Also, given any set A, there is a special function called the identity function on A, and usually denoted i A, which has no effect on its input. That is: i A (x) = x for all x  A. 14

15. Equality of functions If f and g are functions from A to B then f = g if, for every a  A, f(a) = g(a). For example: f, g : R → R f(x) = (x + 1) 2 - 2x g(x) = x 2 + 1 These are equal functions because for any real number x: f(x) = (x + 1) 2 - 2x = x 2 + 2x + 1 - 2x= x 2 + 1= g(x) 15

16. One to One functions A function, f, is called one to one or one-one if, whenever a  b, then f(a)  f(b). That is, for any two distinct inputs of the function, the outputs are also distinct. Put another way, if f(a) = f(b), then necessarily a=b. 16

17. One to One functions Exercise For each of the following functions from {a, b, c, d} to {1, 2, 3, 4}, decide whether it is one to one or not: R 1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R 2 = {(a, 1), (b, 2), (c, 3), (d, 2)} Example The function g : R→R defined by g(x) = 3x + 1 is one-one. 17

18. One to One functions In fact, it is one to one because If g(a) = g(b) then 3a + 1 = 3b + 1, and so 3a = 3b, and finally a = b. On the other hand, the function f : R→R defined by f(x) = x 2 +x+2 is not one to one because f(0) =2 = f(-1) but 0  -1. 18

19. Onto functions A function, g : A → B, is called onto or if, for every b  B there is at least one a  A such that g(a) = b. Exercise For each of the following functions from {a, b, c, d} to {1, 2, 3, 4}, decide whether it is onto or not: R 1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R 2 = {(a, 1), (b, 2), (c, 3), (d, 2)} 19

20. Onto functions A function is onto, if it’s range is equal to its co-domain. Example The function h : R→R defined by h(x) = 5x -2 is onto because, for given b  R we can take a =(b + 2)/5 and then: h(a) = 5a - 2= 5(b + 2)/5 - 2= b + 2 - 2 = b However, f : R → R defined by f(x) = x 2 is not onto because, there is no a  R such that f(a)=-1 20

21. One-One correspondences A function that is both one to one and onto is called a one-one correspondence If f : A → B is a one-one correspondence then, for every b  B there is exactly one a  A such that f(a) = b. The identity function on any set A is always one-one correspondence. 21

22. One to One, Onto, One-One correspondence Let f be a function from A to B f is one to one iff for all x, y ∈ A, if x y, then f(x) f(y) f is onto iff for all y ∈ B, there exists x ∈ A such that f(x) = y f is one-one correspondence iff f is both one to one and onto 22

23. Composition of functions Suppose that f : A → B and g : B → C are functions (the domain of g matches the co-domain of f ) We can string them together, using the output of one rule as input to the next. This operation is known as composition of functions 23

24. Composition of functions Then, given a  A we can take b = f(a) and then c = g(b). This association: a ↦ b ↦ c i.e. a ↦ f(a) ↦ g(f(a)) defines a function h : A → C called the composite or composition of f and g. We write h = g ⃘ f. Very important: the function that is applied second is written first. 24

25. Examples of composition Let f, g : R→R be given by f(x) = 3x - 1 and g(x) = x 2 + 1. Then: (g ⃘ f)(x)= (3x - 1) 2 + 1 The composite of g and f is really obtained by substituting the value of f for the argument of g. Note in this case, we could form the “other” composite: (f ⃘g )(x) = f(g(x)) = f(x 2 + 1)= 3(x 2 + 1) - 1 And that this is not equal to g ⃘ f. 25

26. Rules of composition Let f : A → B, g : B → C and h : C → D be functions. Then: f ⃘ i A = f = i B ⃘ f h ⃘ (g ⃘ f) = (h ⃘ g) ⃘ f Also, if both f and g are one-one then so is g ⃘ f. Likewise, if both are onto so is g ⃘ f. 26

27. Inverse of a function Remember that we can think of a function f : A → B as a relation, that is, as a subset of A × B. Specifically: f = {(a, f(a))  a  A} For a general relation, we can define the notion of its inverse obtained by “flipping” all the ordered pairs belonging to it: f -1 = {(f(a), a)  a  A} 27

28. Exercise For each of the following functions from {a, b, c, d} to {1, 2, 3, 4}, find the inverse relation and decide whether it is a function or not: R 1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R 2 = {(a, 1), (b, 2), (c, 3), (d, 2)} For the function from {a, b, c} to {1, 2, 3, 4}, R 3 = {(a, 1), (b, 2), (c, 3)}, find the inverse relation and decide whether it is a function or not. 28

29. Inverse of a function It seems natural to ask: What conditions, if any, do we need to impose on f in order to ensure that f -1 is a function?. Theorem Let f : A → B be a function. The inverse relation f -1 : B → A is a function if and only if f is a one-to-one correspondence. Note that, if f -1 is a function, then : for all a  A, f -1 (f(a)) = a, and for all b  B, f(f -1 (b)) = b. 29

30. Inverse relations and invertible functions Let R be a relation on A × B The inverse of R, denoted by R −1, is defined as follows: R −1 = {(y, x) : x, y ∈ R} Let f be a function from A to B As f is a relation, we can form the inverse relation f −1 If f −1 is also a function, we say that f is invertible and call f −1 the inverse function 30