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Functions And Relations
 

Functions And Relations

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    Functions And Relations Functions And Relations Presentation Transcript

    • Functions and Relations
      • Objectives
      • 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 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?)
      • 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
    • Exercise 1
    • Exercise 1