Functions
Reading: Chapter 6 (94 – 107)
from the text book
1
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
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
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
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.


1
1
2
3
A
2
3
4
B
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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)
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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.
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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→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 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) = x2+x+2 is not one to one
because
f(0) =2 = f(-1) but 0  -1.
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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)}
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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 always
one-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 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
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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 composite
or composition of f and g. We write h = g ⃘f.
Very important: the function that is applied
second is written first.
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Examples of composition
•
Let f, g : R→R be 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.
25
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 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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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.
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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.
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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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