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Functions

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

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Functions

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

  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

  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 thanoneoutput.

  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 × Bsuch that for allx ∈ A there exists exactly one element y of B such that (x, y) is in the relation

  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).

  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: R1 = {(a, 1), (a, 2), (b, 3), (c, 2)} R2 = {(a, 1), (b, 2), (c, 1)} R3 = {(a, 1), (c, 2)}

  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) = d g(2) = c g(3) = d Read “gof1is equal to d” and so on. This notation is only used for functions and not for more general relations.

  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 1 2 2 3 3 4

  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×Rby the formula: f(x) = x2 + 3

  10. Functions and formulas • To work out the value of the function we just substitute in the formula: f(4) = 42 + 3 = 16 + 3 = 19 • or in another way of writing the same thing (4, 19)  f.We could also write: f = {(x, x2 + 3)  x R}

  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.

  12. Domain, codomain and range • To see the difference between the last two concepts consider f : R →Rdefined by f(x) = x2 • Its co-domain is R but its range is the set of non-negative real numbers.

  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, ftransformsx into f(x) • The domain of f is A • the codomain of f is B • The range of f is {f(x) : x ∈ A}

  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. • Examplef : R→Rdefined 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 iA, which has no effect on its input. That is: iA(x) = x for all x  A.

  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) = x2 + 1 • These are equal functions because for any real number x: f(x) = (x + 1)2 - 2x = x2 + 2x + 1 - 2x= x2 + 1= g(x)

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

  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: R1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R2 = {(a, 1), (b, 2), (c, 3), (d, 2)} Example The function g : R→Rdefined by g(x) = 3x + 1 is one-one.

  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→Rdefined by f(x) = x2+x+2 is not one to one because f(0) =2 = f(-1) but 0  -1.

  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: R1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R2 = {(a, 1), (b, 2), (c, 3), (d, 2)}

  20. Onto functions • A function is onto, if it’s range is equal to its co-domain. Example • The function h : R→Rdefined 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 →Rdefined by f(x) = x2is not onto because, there is noa R such that f(a)=-1

  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.

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

  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

  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.

  25. Examples of composition • Let f, g : R→Rbe given by f(x) = 3x - 1 and g(x) = x2 + 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(x2 + 1)= 3(x2 + 1) - 1 And that this is not equal to g ⃘f.

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

  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}

  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: R1 = {(a, 1), (d, 3), (b, 2), (c, 4)} R2 = {(a, 1), (b, 2), (c, 3), (d, 2)} For the function from {a, b, c} to {1, 2, 3, 4}, R3 = {(a, 1), (b, 2), (c, 3)},find the inverse relation and decide whether it is a function or not.

  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?. TheoremLet 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.

  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

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