| TEACHING MODULE ON RELATION AND FUNCTION |
1. Introduction
Incorporating the recommendation made in the report of National Education commission 1992 and high level National education commission 1998 curriculum. The four unit viz.: Algebra, Vector, Transformation and Statistics are included in new curriculum. In unit algebra, relation, function, surd, polynomial, sequence and series has been included in grade 9 and 10. Among them the topic relation & function has been included in class 9 and 10. According to new syllabus of class 9 and 10 from 2064 BS. Teaching module for teaching relation and function at secondary level has been prepared to teaching the various concepts of relation and function meaningfully and effectively. The content in the module are the content of the optional mathematics class nine and ten respectively.
2. Behavioral Objectives
After completion/study of this teaching module the prospective teachers will be able to
1. Define ordered pair with notation
2. Define Cartesian product with notation
3. Define relation with example
4. Explain the ways of representing relation
5. Define domain and range of relation and find them in given relation.
6. Define function with notation
7. Define domain., co-domain, image, pre-image and range of function and find them in given function.
8. Explain types of function.
9. Explain source simple algebraic functions
10. Define and find composite function
11. Define inverse function and find them in given function.
3. Contents
The following contents are included in the curriculum of grade 9 and 10 to achieve above objective on topic function of optional mathematics.
1. Ordered pair
2. Cartesian product
3. Introduction to relation
4. Ways of representing relation.
5. Domain and range of relation
6. Function
7. Domain, co-domain, image, pre-image, and range of function
8. Types of function
9. Some simple algebraic function
10. Composite function
11. Inverse function
4. Teaching Material.
Flannel boards, and card board, model of various arrow diagram showing relation, function, graph, etc. Material available in daily life situation.
4. Teaching Learning Strategies
Teaching learning models, teaching approaches, methods and teaching learning strategies are synonymously used in everyday speech. The method of instructional materials used for effective teaching and learning called instructional strategy and deductive methods, analytic and synthesis problem solving method; discussion method etc. can be brought into use in teaching as far as possible the teacher should use, students contested co-operative and demonstrative methods. Teaching strategies are the pivot elements of the module and are one of the essential parts of mathematics teaching that influences of the success of mathematics teaching. The following strategies will be fruitful while teaching function and its related problems.
1. Each time check student's pre-requisite knowledge of function and go-ahead with the students learning skills, capacity, age interest and readiness.
2. Try to understand the psychological and physical aspects of student and follow the “learning by doing” procedure as far as possible.
3. Use the mixed teaching strategy based on nature of the lesson that can be helpful for the students to understand easily.
4. Use concrete materials as far as possible and arouse the felling of fair competition with the students.
5. Let them to do move and move practice for a single problem individually in classroom and home work to develop the strong concept on relation and function.
6. Motivate towards the study of set and take care about the level of student’s capacity.
7. Provide right direction to students to prepare a complete note copy of mathematics.
8. Students with different capacity are studying in the class. So try to cover all types of students in an effective chain of learning.
9. Try to find the new ideas, new examples and new way of teaching a new topic on relation and function.
10. Use proper means of evaluation to assess the students progress.
1. Activities
The following paragraph describes in detail the teaching learning activities useful for teaching relation and function.
Activity I: Concept of Ordered Pair
To provide the clear concept of ordered pair to the students teacher should provide various example of a pair of two elements.
For example
Some examples of pairs are (Nepal, Kathmandu), (Sita, Ram), (3,4), (a, b),
(Kathmandu, Nepal)
Then the teacher should ask the following questions.
- Is there different elements in each care (3,4) and(4,3)?
- Are (3,4) and(4,3) same?
With the help of such type of several questions teaching should make it clear that A pair having one element as the first and the other as the second is called an ordered pair. An order pair having ‘a’ as the first element an ‘b’ as second element is denoted by (a, b). An ordered pair (a, b) and (b, a) generally not same. Two ordered a pair
(a, b) and (c, d) are said o be equal If a=c and b=d.
Activity II: concept of Cartesian Product
To give the concept Cartesian product to student teacher should give several examples. For example let us consider any two set A ={1, 2, 3} and B={4,5} are two sets. From above two sets we can make several order pair consisting from each taking first element from A and second from B. Then we have (1, 4), (1, 5), (2,4), 2, 5), (3, 4), (3, 5).
Thus we get the set of collection of order pair {(1, 4), (1, 5), (2, 4), 2, 5), (3, 4), (3, 5)}.
Similarly by giving other many example teacher should make clear concept of Cartesian product and give formal definition as if A and B are two set of all ordered pairs (a, b) s . t. aÎA and b ÎB is called Cartesian product of A and B. and it is denoted by A B. It is read as “A cross B”
In the set- builder notation.
A ×B= {(a, b) / a Î A, b Î B}
Teacher also should give the different ways of representing Cartesian product with the several examples.
Activity III: Concept of Relation
By the example of activity II we have the Cartesian product
A ×B={(1, 4), (1, 5), (2,4), 2, 5), (3, 4), (3, 5)}
Let us take R={(1, 4), (1, 5)). Then teacher should ask following questions to students.
- What is the relation between elements of each pair of R.?
Can we construct other subject A × B?
With the help of such types of several questions. Teacher should make I clear that R is the relation and give the formal definition. Any subset of a Cartesian product A×B is denoted by R,S.g etc.
Relation can be represented by arrow diagrams
Teacher should give other several example for clear concept.
Activity IV: Ways of representing Relation
To provide the clear concept of methods (ways) of representing relation, teacher should give various examples to the students.
Similarly by giving many example teachers should make clear to the students that relation represent various ways.
Activity V: Domain and Range of Relation
To given the clear concept of Domain and Range of relation teacher should provide to students various example. For example
Let A={1, 2, 3} and B={2, 3} then A × B = {(1, 2), (1, 3), (2, 2), (2, 3), (3, 2), (3, 3)}
And R={(1, 2), (2, 3)} by any relation
Then teacher should ask to find two sets by the relation containing as first element and second element of ordered pair respectively. That are C={1, 2}, D={2, 3}.
With the help of several questions. Teacher should make it clear that the set of all first members of the pairs (a, b) of relation R is called the domain of R denoted by dom (R).
Similarly, the set of all second members of the pairs (a,b) of relation R is called the range of R it is denoted by Ran (R).
By the above discussion, Domain and range of relation can be written as in notation form.
Domain = {x|(x, y) ÎR}
range = {y | (x, y) ÎR}
Activity VI : Concept of function
The concept of function under relation should be make clear to the students using various example
For example,
If x = {1, 2, 3} and Y ={2, 3, 4, 5, 6}
Then X × Y = { (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)} here consider sets
R = {(1, 2) (2, 3) (3, 4)}
R1 = {(1, 2) (2, 2) (2, 4)}
Then teacher should ask to student the following question.
Ø What is the different between element of R and R1.
Ø Can we represent R in set builder form
At least teacher should in form to student that R is called function where as R1 is called relation only.
Teacher should other several examples for clear concept of function and teacher should give the formal definition of function as
A function from a set A to a set B is a relation that associates each element of A with a unique elements of b. It is denoted by f: A ® B.
Teacher should also inform by several example that function also can be represent by various ways i.e. graph, table, arrow diagram function machine etc.
Activity VII: Concepts of Domain, Co-domain Pre-image and Range of function
To give the clear concept of above mentioned topic teacher should follow the following example.
Be two function then teacher should as follows question.
- What are the elements of set A in the function f?
- What are the elements of set B in the function f?
- What are the elements of A in function g?
- What are the elements of B in function g? What are the element of B in function g except 5?
- In example (b) what are the element of A which are paired with at least one element of B?
Following several question as above. Teacher should do clear to student that if f: A®B any function then A is called domain of f and is called co-domain of f.
Similarly, if ‘b’ is the unique element of B corresponding to an element ‘a’ of A under the function f, ‘a’ is called the pre-image of ‘b’ under ‘f’ and ‘b’ the image of ‘a’ under f which is written as b= f(a).
Finally teacher should also inform that the subset of B that contains only those elements of B that have pre-images in A is called the range of f. which give obviously range of f Í B.
Teacher should give other many examples to student so that it makes concept clear.
Activity VIII: Types of Function
[onto, into, one-one, many one function]
To give the concept of different types of function teacher should give examples.
Be any two functions then teacher should ask to students following question
Ø What is the co-domain of g?
Ø What is the co-domain of f?
Ø What is the range of g?
Ø What is the range of f?
Ø Are the co-domain of g and range of g are same?
Ø Are the co-domain of f and range of f are same?
With such question teacher should make clear to student that A function f: A®B is said to be onto I co-domain and range of f are same i.e. f(a) =B.
Similarly, in function f, if f(a) Ì B then f is called into function.
(b) To give the concept of one-one function teacher should consider the various examples.
Be any two function
The teacher should ask following question for students
Ø What is the range o X in g ?
Ø What is the co-domain of g?
Ø Is there any different image of pre-image of g?
With the help of the question teacher should inform to students. A function f from A to B is f: A®B is said to be one-one (injection). If distinct element (Pre-image) in A have distinct in image in B.
For clear concept of one-one function teacher should present more example.
(b) Many-one function
To provide the clear concept of many one function teacher should give several example; for example let p and q be any two function give as follows.
Then teacher should ask different question to student
- What are the pre-image of element 4 in function p?
- How many element in pre-image of element 35 in function q?
- How many element in co-domain of p having more than two pre-image?
With the help of these question teacher should inform to the students. If f: A®B be any function then f is called many one function. If two or more then two elements of A has same image in B. for clear concept teacher should give other several example.
Activity IX : Concept of some Simple Algebraic Function [Identity, constant, linear, quadratic, and cubic function.
To give the clear concept of some simple algebraic function teacher should give first the several example of polynomial expression for example.
2, x, x + 2, x2, x2 + 2, x2 + x + 2, x3 , x3 + 2x2 + 2x +2 etc. then teacher should inform that such polynomial expression can be represent as function.
For example
y = f(x) = 2
y = f(x) = x
y = f(x) = x + 2
y = f(x) = x2
y = f(x) = x2 + x + 2
y = f(x) = x3
y = f(x) = x3 + 2x2 + 2x +2
Then teacher should define different algebraic function as following :
y = f(x) = mx +c
where, x Î A and m and c are real constants.
Now teacher should inform to student that
If m=1 and c=0 then f(x)=x thus A function f: A®A defined by y=f(x)=x.
Then f is called identity function. It is denoted by I.
Similarly, If m=0 then y=f(x)=c is called constant function.
Teacher should give other several example to make concept clear.
Again, teacher should inform to student that a function f: A®B defined by
y= f(x) = ax2 +bx + c for xÎA
Where a, b and c are constants is called a quadratic function. i.e. An algebraic function is quadratic function if its degree is 2. And a function f =A®B defined by
y = f(x) = ax3 + bx2 + 2x +d, for xÎ A
Where a, b, c and d are constants is called a cubic function.
i.e. An algebraic function is cubic if its degree is 3.
Teacher should give several examples to make clear concept.
Activity X : Concept of Composite Function
To provide the concept of composite function teacher should give several examples.
For example
Let f : A®B and g: B®C are function defined by:
Here, f gives arrow a to x and
G gives arrow x to 3.
By the notation function
x= f(a) and 3=g(x)
Thus, 3 = g(x) and g(f(a)) x= f(a)
1 = g(y) = g(f(b)) x=f(b)
Thus by above example teacher should provide information that the arrow from A ®c is also as function which is called composite function of g and f.
Activity XI: Concept of Inverse Function
To give the concept of inverse function teacher should first inform to student about inverse image of an element of function by using various example.
By the various example as above, teacher should give f function f: A®B the inverse image of an element y¬b with respect to f is defined as the set of elements in A which have y as their image it is usually denoted by f -1(y) and read as “ f inverse of y”
In symbol if a function is defined by f: A®B then f -1(y) ={ x Î A : y= f(x)}
Teacher should teach this concept by in teaching term method also. By above example teacher should inform to student that when a function f:
A®B is one to one and onto and y be image of A. then f -1(y) = x: xÎA and y= f(x) is called inverse function. It is denoted by f -1:B®A.
Simplify f -1 but f -1 ≠
To make concept clear teacher should give several example to student.
7. Evaluation: (Model Question)
To assessment and test the performance of student following model question can be asked
1. Define the term “ordered pair” with example.
2. Identify which of the following pairs are equal
(a) (1,2) and (2,1) (b) (1,1) and (2,2)
(c) (1,2) and (1,2) (d) (3,3) and (3,3)
3. If the ordered pairs (x+y,1) and (2,2x-y) are equal find x and y.
4. If A={1,2,3} and B={(a, b)} find (a)A ×B (b)B×A (c) A×B=B×A
5. If (2x-1,-3)=)1-,y+3) then find x and y.
6. Let A ={ 1,2,3,4} and B={1,3,5}. Find the relation R from A to B determined by the condition “x<y”.
7. Find the domain and range of the following relation R ={(1,2), (2,4), (3,6), (4,8) }.
8. What is meant by a relation? find the domain and range of a relation
R ={(1,2),(2,3),(3,4),(4,5)}
9. Define function with example.
10. Let f(x)=x+1 be a function defined in the closed interval -1≤x≤1. Find f(-1), f(0), f(1)and f(2)
11. Let A={0,1,2,3,4,5,6} and a function f:A®Q is defined by f(n)=n/2. Find the range of f.
12. Let a function of f: A®A be defined by f(n)=x3 where A ={-1,0,1}. Find the range of the function. Is the function one-one onto or both?
13. What kind of function you know about ? list them.
14. Define onto, into , one –one and many-one function with example.
15. If f(x)= x2+2x+1 then find f(-2) and f(3).
16. If f(x)= 3x+2 then find value of f(0), f(1), f(2), (3).
17. Let f:R®R and g: R®R be defined by f(x)=x+1 and g(x)=x3 find (fof)(x) and (gof)(x).
19. Let f:R®R be defined by f(x)= x2+1, g(x)=x5
find (gof)(x) and (fog)(x).
Bibliography
Bajracharya, B.C. Basic Mathematics Grade XI
Dahal, Karki. Optional Math in Action Grade IX.
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