Functions And Relations

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 i n tersection 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 u nion 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? – We’ll find out later) 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 . {2 n : 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 = x 2 } b {( x, y ): y ≤ x + 1} c {(−2 , −1) , (−1 , −1) , (−1 , 1) , (0 , 1) , (1 , −1)} d {( x, y ): x 2 + y 2 = 1} e {( x, y ): 2 x + 3 y = 6 , x ≥ 0} f {( x, y ): y = 2 x − 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 ) = 2 x 2 + 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( x 2 − 2 x + 1) + ( x − 1) = 2 x 2 − 3 x + 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 ) = 2 x − 4 b g : (−1 , 2] -> R, g ( x ) = x 2

Functions And Relations

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Functions And Relations