2. Functions & Relations An ordered pair, denoted (a,b) is a pair of elements a and b in which a is considered to be the first element and b the second. A relation is a set of ordered pairs. The following are examples of relations:

3. Domain and Range The domain of a relation is the set of all first elements of the ordered pair. The range of a relation is the set of all second elements of the ordered pair. Domain of range of Domain of ; range of

4. Types of relations One to one. Many to one. One to many. Many to many.

5. 5 67895 6789 1234512345      The Rule is ‘ADD 4’

6. Ahmed Peter Ali Jaweria Hamad Paris London Dubai New York Cyprus     Has Visited There are MANY arrows from each person and each place is related to MANY People. It is a MANY to MANY relation.

7. Bilal Peter Salma Alaa George Aziz 62 64 66 Person Has A Mass of Kg       In this case each person has only one mass, yet several people have the same Mass. This is a MANY to ONE relationship

8. One to one: linear, square root graph, hyperbola. Many to many: circle Many to one: parabola, truncus, semicircle. One to many: sideways parabola.

9. FUNCTIONS Many to One Relationship One to One Relationship

10. For a relation to be a function it has to be one to one or many to one relation. Apply THE VERTICAL LINE TEST to test whether a relation is a function. If a vertical line cuts once only anywhere on the graph, the relation is a function. If it cuts more than once, the relation is not a function.

11. (a) (b) (c) (d) (a) and (c)

12. How do we test for one-to- one functions? We apply THE HORIZONTAL LINE TEST. If a horizontal line cuts once only anywhere in the domain, it is a one-to-one function (assuming that it passed the vertical line test for a function first).

13. Which are one-to-one functions? a) b) c) d) a and d

14. Function notation implied or restricted domain all real numbers equation

15. Re-write the following using function notation: Examples 10 &11 on page 182

16. Restricting the domain to create a one-to-one function. Consider the function Draw a graph Apply the horizontal line test Is it a one-to-one function? Restrict the domain of f so that it is one-to-one. HINT: It can be done in more than one way.

17. Not one-to-one

18. Restricted domain, one-to-one now.

19. Another way of restricting the domain to create a one-to-one function. Any other suggestions ?

20. Consider each of the graphs studied so far: linear, quadratic, hyperbola, truncus, square root graph, circle, semicircle. For each apply the vertical line test to determine if it is a function. For each function apply a horizontal line test to determine if it is a one-to-one function.

21. Exercise 5C Q2, 3, 4, 5 Q13 a, c Exercise 5D Q1, 2