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Functions and Relations

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FUNCTIONS AND RELATIONS
Prepared by:
Ms. Ma. Irene G. Gonzales
Learning Objectives
At the end of the period, students should be able to:
• Illustrate a relation and a function;
• Verify if a given relation is a function; and
• Determine dependent and independent variables.
PAIR-ME-UP ACTIVITY
Group A
Ca
K
H
O
Au
Group B
Oxygen
Gold
Calcium
Potassium
Hydrogen
PAIR-ME-UP ACTIVITY
Group A
Manila
Kuala Lumpur
Jakarta
Hanoi
Bangkok
Group B
Indonesia
Malaysia
Philippines
Thailand
Vietnam
What is a Relation?
• It involves the association of an individual or object with
another individual or object.
Learning Objective: Students should be able to illustrate a relation and a function.
What is a Relation?
• It involves the association of an individual or object with
another individual or object.
Learning Objective: Students should be able to illustrate a relation and a function.

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Functions and Relations

  • 1. FUNCTIONS AND RELATIONS Prepared by: Ms. Ma. Irene G. Gonzales
  • 2. Learning Objectives At the end of the period, students should be able to: • Illustrate a relation and a function; • Verify if a given relation is a function; and • Determine dependent and independent variables.
  • 3. PAIR-ME-UP ACTIVITY Group A Ca K H O Au Group B Oxygen Gold Calcium Potassium Hydrogen
  • 4. PAIR-ME-UP ACTIVITY Group A Manila Kuala Lumpur Jakarta Hanoi Bangkok Group B Indonesia Malaysia Philippines Thailand Vietnam
  • 5. What is a Relation? • It involves the association of an individual or object with another individual or object. Learning Objective: Students should be able to illustrate a relation and a function.
  • 6. What is a Relation? • It involves the association of an individual or object with another individual or object. Learning Objective: Students should be able to illustrate a relation and a function.
  • 7. Mathematically speaking… • A relation is any set of one or more ordered pairs. • There is one value of y that corresponds to EVERY value of x. – DOMAIN: set of first coordinates – RANGE: set of second coordinates Learning Objective: Students should be able to illustrate a relation and a function.
  • 8. Relation as a Set of Ordered Pairs • Using the elements of A = {1, 2, 3, 4, 5}, construct the set of ordered pairs F that describes the relation “The abscissa is one greater than the ordinate.” • Solution F = {(2,1), (3,2), (4,3), (5,4)} Learning Objective: Students should be able to illustrate a relation and a function.
  • 9. Relation described by a Table Filipino Heroes Date of Birth Jose Rizal June 19, 1861 Andres Bonifacio November 30, 1863 Apolinario Mabini July 23, 1864 Emilio Aguinaldo March 22, 1869 Gabriela Silang March 19, 1731 Learning Objective: Students should be able to illustrate a relation and a function.
  • 10. Relation Expressed in Arrow Diagrams Learning Objective: Students should be able to illustrate a relation and a function. Manila Kuala Lumpur Jakarta Hanoi Bangkok Indonesia Malaysia Philippines Thailand Vietnam D R
  • 11. One-to-many Learning Objective: Students should be able to illustrate a relation and a function. 1 4 9 -3 -2 -1 1 2 3 X Y
  • 12. Many-to-one Learning Objective: Students should be able to illustrate a relation and a function. -3 -2 -1 1 2 3 2 4 6 X Y
  • 13. Relation as an Equation or Formula Learning Objective: Students should be able to determine dependent and independent variables. • For instance, the equation y = x + 1 • Between the two variables, which one is the independent variable? Which is the dependent variable?
  • 14. Relation as an Equation or Formula Learning Objective: Students should be able to determine dependent and independent variables. • For instance, the equation y = 4x + 1 • Between the two variables, which one is the independent variable? Which is the dependent variable?
  • 15. Relation Described Graphically Learning Objective: Students should be able to illustrate a relation and a function.
  • 16. Differentiate a function from a relation… • A special type of relation illustrated by the following sets of ordered pairs: A = {(1,2), (3,4), (5,6), (7,8)} B = {(-8,7), (-2,3), (-4,5)} C = {(4,3), (5,8), (7,9), (-1,2)} The following sets are relations but not functions: D = {(1,2), (1,3), (1,4), (1,5)} E = {(-8,1), (-8,2), (4, 3), (4, 5)} Learning Objective: Students should be able to verify if a given relation is a function.
  • 17. What is a FUNCTION? • It is a relation in which each element of the domain corresponds to exactly one element of the range. • In simple terms, it is a special type of relation wherein no two ordered pairs have the same abscissa. • In other words, “x is faithful to EXACTLY one y” Learning Objective: Students should be able to verify if a given relation is a function.
  • 18. Real-Life Relationships of Functions H = {(Ms. Irene, August), (Joseph, March), (Clyle, November), (Tim, December)} J = {(Mr. Perilla, St. Benedict), (Mr. Guevara, St. Bertha), (Ms. Gonzales, St. Barbara), (Mr. Ulaye, St. Bernard)} Learning Objective: Students should be able to verify if a given relation is a function.
  • 19. Characteristics of a Function 1. Each element in domain X must be matched with EXACTLY one element in range Y. 2. Some elements in range Y may not be matched with any element in domain X. 3. Two or more elements in domain X may be matched with the same element in range Y. Learning Objective: Students should be able to verify if a given relation is a function.
  • 20. Vertical Line Test A graph represents a FUNCTION if and only if no vertical line intersects the graph in more than one point. Learning Objective: Students should be able to verify if a given relation is a function.
  • 21. One-to-many Learning Objective: Students should be able to illustrate a relation and a function. 1 4 9 -3 -2 -1 1 2 3 X Y
  • 22. Many-to-one Learning Objective: Students should be able to illustrate a relation and a function. -3 -2 -1 1 2 3 2 4 6 X Y
  • 23. Function or not a function? Learning Objective: Students should be able to illustrate a relation and a function. 12 5 8 2 4 X Y
  • 24. Function or not a function? Learning Objective: Students should be able to illustrate a relation and a function. 12 5 8 2 4 X Y
  • 25. Function or not a function? Learning Objective: Students should be able to illustrate a relation and a function. 12 5 8 2 4 6 X Y
  • 26. Function or not a function? Learning Objective: Students should be able to illustrate a relation and a function. 12 5 8 2 4 6 X Y
  • 27. Get ready for the Learning Activity 4.3