What are mathematical relations and functions? Do Now: Describe relations and functions in your life.

What is a relation?  A relation is any set of ordered pairs.  These pairs can be numbers, names, words, etc.  A relation has a domain and a range  The domain is the set of independent elements, the set of x-values one can plug in (Input)  The range is the set of dependent elements, the set of y-values that come out (Output)  A relation is any set of ordered pairs.  These pairs can be numbers, names, words, etc.  A relation has a domain and a range  The domain is the set of independent elements, the set of x-values one can plug in (Input)  The range is the set of dependent elements, the set of y-values that come out (Output)

What are the Domain and Range?  Relation A: {(February, 2), (April, 4), (June, 6), (August, 8), (October, 10)}  Relation B: {(1991, 28), (1996, 29), (1997, 28), (2000, 29), (2003, 28)}  Relation C: { (4, 13), (-2, 7), (5, 14), (-8, 1), (-4, 5)}  Since we do not know the EXACT relation between the Domain and Range, we can not add other pairs to this lists.  Relation A: {(February, 2), (April, 4), (June, 6), (August, 8), (October, 10)}  Relation B: {(1991, 28), (1996, 29), (1997, 28), (2000, 29), (2003, 28)}  Relation C: { (4, 13), (-2, 7), (5, 14), (-8, 1), (-4, 5)}  Since we do not know the EXACT relation between the Domain and Range, we can not add other pairs to this lists.

Try on your own  Pages 138-9

What is a function?  A function is a relation in which each element of the domain corresponds to a UNIQUE element in the range.  Put another way, the function maps each element of the domain to only ONE element in the range.  When you plug in an x, you only get one y back out.  A function is a relation in which each element of the domain corresponds to a UNIQUE element in the range.  Put another way, the function maps each element of the domain to only ONE element in the range.  When you plug in an x, you only get one y back out.

Explain if the following are functions?  A: {(purple, lilac), (yellow, daffodil), (pink, carnation), (purple, tulip)}  B: {(-7, 3), (-3, 8), (-1, 10), (4, 3)}  C: {(4, 2), (9, -3), (25, 5), (16, -4), (9,3)}  D: { (1, 13), (5, 10), (9, -7), (13, -4), (17, -1)}  A: {(purple, lilac), (yellow, daffodil), (pink, carnation), (purple, tulip)}  B: {(-7, 3), (-3, 8), (-1, 10), (4, 3)}  C: {(4, 2), (9, -3), (25, 5), (16, -4), (9,3)}  D: { (1, 13), (5, 10), (9, -7), (13, -4), (17, -1)}

Are there other ways to see if a relation is a function?  If we have a visual representation of the relation, also known as a graph, then there is a simple test we can perform.  Vertical line test  If we can draw a vertical line anywhere on the graph and it intersects the graph at only one point, then the relation is a function.  If we have a visual representation of the relation, also known as a graph, then there is a simple test we can perform.  Vertical line test  If we can draw a vertical line anywhere on the graph and it intersects the graph at only one point, then the relation is a function.

How do we use functions in our lives?  Any time that we can represent a situation as a graph with one dependent variable (y-axis) that changes based on the independent variable (x-axis), then we can use a function to describe the relation.  Distance and time  Wages and education  Grades and study time  Remember: Independent is on the horizontal (x-axis), dependent is on the vertical (y-axis).  Any time that we can represent a situation as a graph with one dependent variable (y-axis) that changes based on the independent variable (x-axis), then we can use a function to describe the relation.  Distance and time  Wages and education  Grades and study time  Remember: Independent is on the horizontal (x-axis), dependent is on the vertical (y-axis).

Example  A New York subway train slows down as it approaches the 66th St. station, stops at the station for 2 minutes, and then continues on its route. How could we draw a graph of this function?

Describe the Function  You are hiking in Glacier National Park and you turn a corner and run into some bears. Your heart starts pounding, but as you walk away you get it under control. Draw the graph of a function that compares your heartbeat (y) to time (x).

Summary/HW  Describe a MATHEMATICAL function you have experienced.  HW: pg #1-18 even  Describe a MATHEMATICAL function you have experienced.  HW: pg #1-18 even