U1.1 Dependency Relationships PAPA2

1.2 Vocabulary Relation Continuous Function Interval Notation Domain Function Notation Range Infinity Independent Variable Dependent Variable Mapping Vertical line test Discrete

Definitions… Relation – a set of ordered pairs that represent a mapping/pairing of input values with output values Function – a special type of relation in which each x value is paired with exactly one y value Domain – the set of all x values in a relation Range – the set of all y values in a relation Independent Variable: The input variable (in most cases x). Dependent Variable: The output variable (in most cases y).

U1.1 Dependency Relationships – Explore

Problem Situation 1: You got a job at the movie theater earning $8 Problem Situation 1: You got a job at the movie theater earning $8.25 a hour.  Problem situation 1:  What is the independent variable? What is the dependent variable?  We can then say “The ________________________ depends on __________________________.”  What is the domain for this relation?  What is the range for the relation?  Is this relationship a function?  Is this situation a continuous or discrete relationship?

Problem Situation 2: The Band booster club is selling WHS Salsa to raise money. They sell each Salsa container for $10.00 each.  Problem situation 2:  What is the independent variable? What is the dependent variable?  We can then say “The ________________________ depends on __________________________.”  What is the domain for this relation?  What is the range for the relation?  Is this relationship a function?  Is this situation a continuous or discrete relationship?

Identifying Functions If any of the x values repeat, then the relation is NOT a function. Example 1: Is the relationship in the table a function? X -2 4 6 5 7 y -3 1 It is a function! Why? X-values don’t repeat

Mapping Relations A process of matching the elements of the domain (x) with the elements of the range (y). Remember that in order to be a function a member of the domain (x) can map to ONLY one member of the range (y).

Real-Life Mapping Example 2 Is it possible for Brandon and Luis to both have Chemistry 1st period? Yes Is it possible for Isaac to have 2 classes during A1? No. He doesn’t know where to go. Why is Brittney confused? She doesn’t know where to go. Point being,… the x-values need to “know where to go.” Different x-values can go to the same y-value as long as they “know where to go.” Assume that no student below has “late-arrival.” Student (x) A1 Class (y) Rose Health Brandon Spanish Luis Chemistry Monica Algebra 2 Isaac English 2 Brittney

Are the following functions? Yes. Why? Yes. Why? There is only one arrow coming from each x. There is only one arrow coming from each x.

Are the following functions? No. Why? No. Why? Domain is paired twice! One does not have a pair

Identifying Functions The vertical line test – If you move a vertical line across the graph and the graph only touches the line in one place, then it passes the test, and it is a function. Is a vertical line a function? No Why? All x values are the same.

Is it a function? The vertical (Red) line does not touch the graphed line more than one time at any given place on the graph. It is a function!

Is it a function? The vertical line touches the graph in two places at many points on the graph. It is not a function! It fails the vertical line test.

New Terms… Discrete – it has a countable number of possible values Example 3 Discrete – it has a countable number of possible values Continuous – it takes on all values in an interval of numbers Example 4

Discrete/ Continuous Examples

Examples 5 & 6 Find the domain, range, and state whether or not it is a function. 5. {(6, 3), (2, -1), (4,8), (-2, -1)} D:{-2, 2, 4, 6} R:{-1, 3, 8} function 6. The set of values such that x is all real numbers equal or between negative 2 and 2. The set of values such that y is all real numbers equal or between negative 2 and 2.

Function Notation F(x) x Find… (x,f(x))

Example 7 Find

Example 8 Find

Practice 1… Find

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Practice 2… Find

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Dependency Relationships Part 2

Interval Notation & The Number Line Symbols To Know… ( ) Meaning “not including the end point” [ ] Meaning “including the end point” U Union Symbol (Or) +∞ Positive Infinity -∞ Negative Infinity

Example 9 How would you represent the number line with interval notation? … -5 – 4 -3 -2 -1 0 1 2 3 4 5 6 ... ○ (-3,5) The interval represents the real numbers between –3 and 5 but does not include the -3 or the 5 as part of the set of numbers. The smaller of the two numbers is always on the left while the larger is always on the right -- just like on the number line, the smaller number is on the left and the larger number is on the right. The parentheses indicate that the beginning and the ending numbers do not belong to the interval.

Example 10 How would you show that the end numbers belong to the set in the graph below. … -5 – 4 -3 -2 -1 0 1 2 3 4 5 6 … ● What are the numbers being represented by this number line? all real numbers between and including -1 and 2. Remember that real numbers are more than just integers. What symbol would we use in interval notation to represent the included endpoints? Bracket indicate that the endpoint is included in the set of numbers. What is the interval? [-1,2]

Example 11 Use interval notation to describe the set of numbers shown on the number line below: (-1,1] Is -1 included in the set? No … -5 – 4 -3 -2 -1 0 1 2 3 4 5 6 … ( ]

Example 12 Use interval notation to describe the set of numbers shown on the number line below: Two groups of numbers are represented on the number line. What symbol do we use to show that we have to intervals? The union “or” symbol U (- 4,-2] U [2,4) Is -1 included in the set? Why? No, -1 falls between the two intervals of numbers. … -5 – 4 -3 -2 -1 0 1 2 3 4 5 6 … ( ]

Example 13 State the interval. Describe all real numbers larger than –2 and smaller than 4 using interval notation. (-2,4) Use interval notation to describe all numbers larger than or equal to –3.

Can you … Describe legal driving ages using interval notation… assuming you can start driving legally at age 16 and you must stop driving for health reasons at age 80. [16,80]

Absent Students Remember: You are responsible for collecting ALL handouts for the day you are out, in addition to copying the notes.