Week 5 – SS1 Second Term Further Mathematics Notes

WEEK FIVE                                                  DATE…………… MAPPING AND FUNCTIONS

• Concept of mapping and function
• Domain, Co-domain of function –     Types of mapping.

MAPPING

Definition, Concept, Example and evaluation.
Definition: This is the rule which assign an element x in set A to another unique element y in set B.
The set A is called the Domain while set B is the Co- domain

Image: This is the unique element in set B produced by an element in set A.
Range: This is the collection of all the images of the elements of the domain.

  Using the diagram above: f(w)= g, f(x)= b, f(y)=f, f(z)=a a, b, f and g are the images of elements a,b,c and d respectively. Range = {a, b, f, g,}

  The rule which associates each element in set A to a unique element in set B is denoted by any of the following notations: f : A → B or f: A→ B

 
 
 
 FUNCTION: A function is a mapping whose co-domain is the set of numbers.
X F Y

 
 
 
 
 
 
 
 
 
 
 
  Therefore, f (10) =4, f (9) =3 e.t.c

 Example 1: Given f(x) = 3x² + 2, find the values of (a) f (4) (b) f (-3) (c) f (-1/2)

 SOLUTION:
F(x) = 3x²+ 2

1. F(4), i.e x=4

F(4) = 3(4²) + 2 = 3(16) + 2
= 48 + 2
= 50

1. F(-3) = 3(-3)²+ 2

= 3(9) +2 = 27 +2
= 29

1. F(-1/2) = 3(-1/2)²+ 2

= 3(1/4) + 2 = 3 + 2
4
=11/4.
Example 2: Determine the domain D of the mapping, g:x→ 2x² – 1, if R= { 1,7,17} is the range and g is defined on D. SOLUTION:
g(x) = 2x²– 1, R = {1,7,17} To find the domain, when g(x) = 1, 1= 2x² -1 1+1 = 2x²
x² = 2/2 x=1
When g(x) = 7,
7 = 2x²-1
7+1 = 2x² 8 =2x²
x²= 4, x= 2
When g(x)     = 17,

1. =x²-1

17+1 = 2x²

1. =x²

x² = 9, x= 3

 Domain D ={1, 2,3}

EVALUATION

1. Given f(x) = x²+ 4x +3 find the values of.

    (a) f(2) (b) f(½) (c) f(-3)

1. Given that f(x) = ax + b and that f(2) = 7 ,f(3) = 12. Find a and b.

TYPES OF MAPPING

One-One mapping: A mapping is one-one if different elements in the domain have different images in the co- domain. If x₁= x_{2 then} f(x₁) = f(x₂)

  A 2X+3 B
-12
-51
3 4

Onto Mapping: A mapping is onto if every element of the co- domain is at least an image of elements in the domain. E.g Let A = {-1, 0, 1} f : A → A be a mapping defined by f(x)= x³.

  A F=X³ B

    
The mapping is onto and one-one.
NB: In an onto mapping, the range is the same as the co- domain.
Identity Mapping: This is a mapping which takes an element onto itself. If f: x→ x is a mapping such that f(x) = x for all x € X.

      
X X

The mapping is one –one and onto. It has a unique property that the domain, the co-domain and the range are equal.

 Constant Mapping: This is the mapping which assigns every element in the domain to the same image in the co- domain.
X Y

The range of a constant mapping consists of only one element.

 Important Notes:
1.
X F Y

The relation F above is not a mapping because element q in X has no image in Y.

 
 2.
A g B

 The relation is not a mapping because element z in the domain has two images in the co –domain.

EVALUATION

1.     Given the mapping diagram below:
X P Y

 
 
 
 
  (a). Determine the rule of the mapping

1. Is the mapping one- one? Is it onto?
2. What is the range of the mapping?

GENERAL EVALUATION

1. Solve the system of equation; 2^{x + y} =32, 3^{3y – x} = 27
2. Given h(x) = x³-6x² – 3x +5 find the values of.

    (a) h(-2) (b) h(-½) (c) h(3)

1. Given that g(x) = 2p – q and that g(2) = 20 ,g(-3) = 15. Find p and q.
2. Given the functions h(y) = 3y² –y+5, p(y) = 6y³ + 7y²+7y+15. Simplify, as far as possible, the expressions

(a) 3h(y) – p(y) (b) h(y) p(y) (c) h(y)/p(y)

 READING ASSIGNMENT: Read Mapping, Further Mathematics Project 2, and page 25- 35.

WEEKEND ASSIGNMENT

1. If every element in the domain have different image I the co-domain, such type of mapping is called ——-

(a) constant mapping (b) onto mapping (c) one- to – one mapping

1. A mapping f is called —— if every element of the co-domain is an image of at least one element in the domain (a) constant mapping (b) onto mapping (c) one- to – one mapping 3. Given f(y) = p^x and f (3) = 81, determine the value of x.

A -4 B 27 C 4
4 The rule that assign an element to two or non-empty set is (a) logic (b) set (c) mapping 5 If f is a function defined by f(x) = 2x² – 3, find f(-3). A. -15 B 18 C. 15
    

    THEORY

1. Determine the domain D of the mapping f: x 2x – 2, if c = { -3, -1, 5 } is range and f is defined on D
2. Given that h: x x² + 2x – 3 is a mapping defined on the set A = { -1, 0, 1, 2}. Find the range of h.