Functions

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

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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

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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.

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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

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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).

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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)} 6

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 “g of 1 is equal to d” and so on. This notation is only used for functions and not for more general relations.

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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

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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) = x2 + 3

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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}10

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.

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Domain, codomain and range

• To see the difference between the last two concepts consider f : R → R defined by

f(x) = x2

• Its co-domain is R but its range is the set of non-negative real numbers.

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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}∈

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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 iA, which has no effect on its input. That is:

iA(x) = x for all x A.

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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)15

One to One functions

• A function, f, is called one to one or one-oneif, 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.

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One to One functions

ExerciseFor 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→R defined by g(x) = 3x + 1 is one-one.

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One to One functions

• In fact, it is one to one becauseIf 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) = x2+x+2 is not one to one because

f(0) =2 = f(-1) but 0 -1.18

Onto functions

• A function, g : A → B, is called onto orif, for every b B there is at least one a A such that g(a) = b.

ExerciseFor 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)}19

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) = x2 is not

onto because, there is no a R such that f(a)=-1

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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 alwaysone-one correspondence.

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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

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Composition of functions

• Suppose that f : A → B and g : B → C are functions (the domain of g matchesthe 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

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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 compositeor composition of f and g. We write h = g ⃘f.Very important: the function that is applied second is written first. 24

Examples of composition

• Let f, g : R→R be given by f(x) = 3x - 1 andg(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) - 1And that this is not equal to g ⃘f.

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Rules of composition

• Let f : A → B, g : B → C and h : C → D be functions. Then:

f ⃘iA = f = iB ⃘fh ⃘(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.

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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}

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ExerciseFor 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.

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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

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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