1. CHAPTER I Piecewise Functions and Function Operations
Dr. Rebecca C. Tolentino
Pamantasan ng Lungsod ng Maynila

2. Lesson 1: Functions Learning Outcomes:
At the end of the lesson, the learner is able to define and differentiate relations and functions, and represent real life situations using functions.

3. Definition
A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y, and write xy or as the ordered pair (x,y).
Sullivan, College Algebra

4. Definition: Relation
A relation is a correspondence between two sets ( called the domain and the range) such that to each element of the domain, there is assigned one or more elements of the range.
To define a relation three things must be designated: 
the domain set,
the range set and
the rule of assignment.

5. Example (2,3), (2,4), (3,7), and (5,2) defines a relation with
        Domain:  {2,3,5}
        Range:  {2,3,4,7}

6. Mapping Diagram

7. Graph

8. Illustration: Machine
Machine with an input and an output, such that the output is related to the input by some rule.

9. Machine A INPUT: ANY WHOLE NUMBER FROM 1 TO 10
OUTPUT: THE INPUT NUMBER MULTIPLIED BY 2
OUTPUT

10. INPUT: ANY LETTER FROM THE ALPHABET
Machine B
INPUT
INPUT: ANY LETTER FROM THE ALPHABET
OUTPUT: IF VOWEL, 5;
IF CONSONANT, 9
OUTPUT

11. IF GREATER THAN 5, B; ELSE, C
Machine C
INPUT
INPUT: DIGITS 0 TO 9
OUTPUT: IF EVEN, A;
IF GREATER THAN 5, B; ELSE, C
OUTPUT

12. Real Life Examples of Relation
Consider the relation that sends a student to that student's age.
Consider the relation that sends a student to the courses that student is taking.
Consider the relation that sends a parent to the parent's child.
Consider the relation that sends a key word either to its matches from the Yahoo search engine or to the statement "No matches found."

13. Definition: Function
A function is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned exactly one element of the range.

14. A function is a relation where each element in the domain is related to only one value in the range by some rule.

15. Exercise:
Which of the machines in the previous slides illustrate a function?

16. Which of the following mapping diagrams represent functions?b c x y z a b x y z a b c x Y a b c x y z

17. Another Definition: Function
A function is a set of ordered pairs (x,y) such that no two ordered pairs have the same x-value but different y values.
Example:
f={(1,1),(2,4),(3,9),(4,16)}
Domain = {1,2,3,4}
Range={1,4,9,16}
Rule: f(x)=x2

18. Which of the following relations are functions?

19. Vertical Line Test
If a vertical line intersects the graph in two or more points, then this means that there is an x-value corresponding to two or more y-values, and therefore, the graph does not represent a function.
Example:

20. Which of the following are graphs of functions?b c

21. Which of the following equations are functions?

22. Which of the following real life examples illustrate functions?
Consider the relation that sends a student to that student's age.
Consider the relation that sends a student to the courses that student is taking.
Consider the relation that sends a parent to the parent's child.
Consider the relation that sends a key word either to its matches from the Yahoo search engine or to the statement "No matches found."

23. Definition: Domain and Range
– set of all first elements or set of all values of x
-set of permissible inputs
Range
– set of all second elements or the set of all values of y
-set of all resulting output

24. Example: Identify the domain and range of each relation on the previous slides.
Ordered Pairs in slide 18
Graphs in slide 20
Equations in slide 21
Real life situations in slide

25. Identify the domain using set-builder notation.
{x: x , x≥1}
{x: x , -1≤x ≤ 1}
{x: x }
{x: x , x ≠ 1}

26. Group Activity 1
Provide examples of functions and ordinary relations using the following :
mapping diagrams,
set of ordered pairs,
graphs
machines,
equations.

27. Format of Output

28. For questions or suggestions:
For questions or suggestions: