Functions And Relations

Functions and RelationsObjectives•To understand and use the notation of sets, including the symbols ∈, ⊆, ∩, ∪, ∅ and \.•To use the notation for sets of numbers.•To understand the concept of relation.•To understand the terms domain and range.•To understand the concept of function.•To understand the term one-to-one.•To understand the terms implied domain, restriction of a function, hybrid function, and odd and even functions.•To understand the modulus function.•To understand and use sums and products of functions.•To define composite functions.•To understand and find inverse functions.•To apply a knowledge of functions to solving problems.

Set Notation• A set is a collection of objects e.g A = {3,4}.• The objects in the set are known as the elements or

members of the set. • For example, you are ‘elements’ of our class ‘set’.• 3 ∈ A means ‘3 is a member of set A’ or ‘3 belongs to A’.• 6 ∉ A means ‘6 is not an element of A’.

Set Notation• If x ∈ B implies x ∈ A, then B is a subset of A, we write B ⊆ A.

This expression can also be read as ‘B is contained in A’ or ‘A contains B’.

• The set ∅ is called the empty set or null set. • A ∩ B is called the intersection of A and B . Thus x ∈ A ∩ B if

and only if x ∈ A and x ∈ B.• A ∩ B = ∅ if the sets A and B have no elements in common.• A ∪ B , is the union of A and B. If elements are in both A and

B they are only included in the union once.• The set difference of two sets A and B is denoted A\B (A but

not B)• Example 1: A = {1, 2, 3, 7}; B = {3, 4, 5, 6, 7}

Find: a) A ∩ B b) A ∪ B c) A\B d) B\A • Solution: a) A ∩ B = {3, 7}

b) A ∪ B = {1, 2, 3, 4, 5, 6, 7} c) A\B = {1, 2} d) B\A = {4, 5, 6}

Sets of numbers

• N: Natural numbers {1, 2, 3, 4, . . .} • Z: Integers {. . . ,−2,−1, 0, 1, 2, . . .}• Q: Rational numbers – can be written as a fraction. Each

rational number may be written as a terminating or recurring decimal.

• The real numbers that are not rational numbers are called irrational (e.g. π and √2).

• R: Real numbers. (How can a number not be real?)• It is clear that N ⊆ Z ⊆ Q ⊆ R and this may be represented

by the diagram:

Sets of numbers• The following are also subsets of the real numbers for

which there are special notations:• R+ = {x: x > 0}• R− = {x: x < 0}• R\{0} is the set of real numbers excluding 0.• Z+ = {x: x ∈ Z, x > 0}

Note: • {x: 0 < x < 1} is the set of all real numbers between 0 and

1.• {x: x > 0, x rational} is the set of all positive rational

numbers.• {2n: n = 0, 1, 2, . . .} is the set of all even numbers.

Representing sets of numbers on a number line

• Among the most important subsets of R are the intervals. • (-2, 4) means all ‘real’ numbers between (but not including) -

2 and 4.• [3, 7] means all ‘real’ numbers between 3 and 7 inclusive.• [4, ∞) means all ‘real’ numbers greater than or equal to 4.• (-∞, 3) means all ‘real’ numbers less than 3.

Representing sets of numbers on a number line

• Example 2: Illustrate each of the following intervals of the real numbers on a number line:

• a [−2, 3] b (−3, 4] c (−∞, 5] d (−2, 4) e (−3,∞)

Describing relations and functions• An ordered pair, denoted (x, y), is a pair of elements x and y

in which x is considered to be the first element and y the second (it doesn’t mean they have to be in numerical order).

• A relation is a set of ordered pairs. The following are examples of relations:

• S = {(1, 1), (1, 2), (3, 4), (5, 6)}• T = {(−3, 5), (4, 12), (5, 12), (7,−6)}• The domain of a relation S is the set of all first elements of the

ordered pairs in S.• The range of a relation S is the set of all second elements of

the ordered pairs in S.• In the above examples:

domain of S = {1, 3, 5}; range of S = {1, 2, 4, 6}domain of T = {−3, 4, 5, 7}; range of T = {5, 12, −6}

• A relation may be defined by a rule which pairs the elements in its domain and range.

• Let’s watch an example.

Describing relations and functions• Example 3: Sketch the graph of

each of the following relations and state the domain and range of each.

a {(x, y): y = x2} b {(x, y): y ≤ x + 1}c {(−2,−1), (−1,−1), (−1, 1), (0, 1),

(1,−1)} d {(x, y): x2 + y2 = 1}e {(x, y): 2x + 3y = 6, x ≥ 0} f {(x, y): y = 2x − 1, x ∈ [−1, 2]}

Describing relations and functions• A function is a relation such that no

two ordered pairs of the relation have the same first element.

• For instance, in Example 3, a, e and f are functions but b, c and d are not.

• Functions are usually denoted by lower case letters such as f, g, h.

• The definition of a function tells us that for each x in the domain of f there is a unique element, y, in the range.

• The element y is denoted by f(x) (read ‘f of x’).

Describing relations and functions• Example 4: If f (x) = 2x2 + x, find f (3), f (−2) and f (x − 1).• Solution• f (3) = 2(3)2 + 3 = 21• f (−2) = 2(−2)2 − 2 = 6• f (x − 1) = 2(x − 1)2 + x − 1• = 2(x2 − 2x + 1) + (x − 1)• = 2x2 − 3x + 1

Describing relations and functions• Example 5: For each of the following, sketch the graph and

state the range:• a f : [−2, 4] → R, f (x) = 2x − 4 • b g : (−1, 2] → R, g(x) = x2

Exercise 1

Exercise 1