CHAPTER I Piecewise Functions and Function Operations Dr. Rebecca C. Tolentino Pamantasan ng Lungsod ng Maynila
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.
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 xy or as the ordered pair (x,y). Sullivan, College Algebra
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.
Example (2,3), (2,4), (3,7), and (5,2) defines a relation with Domain: {2,3,5} Range: {2,3,4,7}
Mapping Diagram
Graph
Illustration: Machine Machine with an input and an output, such that the output is related to the input by some rule.
Machine A INPUT: ANY WHOLE NUMBER FROM 1 TO 10 OUTPUT: THE INPUT NUMBER MULTIPLIED BY 2 OUTPUT
INPUT: ANY LETTER FROM THE ALPHABET Machine B INPUT INPUT: ANY LETTER FROM THE ALPHABET OUTPUT: IF VOWEL, 5; IF CONSONANT, 9 OUTPUT
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
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."
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.
A function is a relation where each element in the domain is related to only one value in the range by some rule.
Exercise: Which of the machines in the previous slides illustrate a function?
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
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
Which of the following relations are functions?
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:
Which of the following are graphs of functions? b c
Which of the following equations are functions?
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."
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
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
Identify the domain using set-builder notation. {x: x , x≥1} {x: x , -1≤x ≤ 1} {x: x } {x: x , x ≠ 1}
Group Activity 1 Provide examples of functions and ordinary relations using the following : mapping diagrams, set of ordered pairs, graphs machines, equations.
Format of Output
For questions or suggestions: rctolentino@plm.edu.ph For questions or suggestions: