Introduction to Relations and Functions

MCR3U/MHF4U – Lesson #1 Introduction to Relations and Functions

 Mathematics is a study of patterns, order and how different thing relate to each other.

 This year is grade 11 this course will focus on how humans can use math to relate to concepts to each other. We use math in this way so that we can understand natural processes, make predictions and plan ahead.

 To start we first examine what are known as single variable relations

Definition 1.1: In mathematics a relation is a rule, pattern or formula that allows one to figure out one set of data from another.

Ex. #1: A concert is held at the Sky Dome, each ticket is $5. Tickets Sold Revenue 3 15 **Define 4 20 Revenue** 5 25 10 50 50 250  Here each ticket is worth $5 and your revenue depends directly on how many tickets you sell. This is a relation because you can figure out your revenue by multiplying your number of tickets by 5.

Ex. #2: Shoe Size Height (cm) 7.5 161 10 158 11.5 175 8.5 177  Is this a relation?

 Looking at the tickets example above there are two variables. The number of tickets sold and the amount of revenue Definition 1.2: A variable is a quantity that can become any value out of a set of different values. In mathematics a variable is often represented by a letter ( x ).

 In the case above the number of tickets is a variable. This variable can assume any whole number value. For example it could be 3, 50 or 100.  However the number of tickets sold cannot be a negative value or be more than the number of people the room or place can hold.

 Now after the concert at the Sky Dome how can we tell how much money was brought in?

Definition 1.3: A dependent variable is a variable whose value depends on the value of another variable. For example, the concert revenue depends on the number of tickets sold.

Definition 1.4: An independent variable is a variable whose value is arbitrary (can be chosen). An independent variable is the variable whose value does not rely on anything else.

 For the concert our number of tickets sold is the independent variable while the amount of revenue is our dependent variable because the revenue depends on the tickets sold. §1.1 - Functions:

Definition 1.4: A function is any relation that for each individual independent variable there is only ONE corresponding dependent variable.

Ex. #3 – The Ideal Vending Machine:

 A vending machine allows you to put in money and in return out comes a select food or drink depending on how much money was put in.

 This is a function. One starts with a select amount of money (the independent variable). The machine is then the rule or formula. The takes in the money and button press, parts move and then a drink is dispersed. The drink is the dependent variable since the type of drink relies on the amount of money put in.

Independent Variable (Amount of Money)

The Food or Drink (Dependent Variable) The Machine (Rule or Formula)

 The independent variable in a function, just like a vending machine, can take on set of values. These set of values is called a function’s domain. Definition 1.5: The Domain of a function is the possible values the independent variable, of a function, can be.

Ex. #4 – The domain of the vending machine function is the number of different prices listed on the machine (ie. $1.50, $1.75, $2.00 …).

Definition 1.6: The range of a function is all the possible values that the dependent variable can take on.

Ex. #5 – The range of the vending machine function is the number of different foods or drinks one can get.

 REMEMBER something is ONLY a function if you can go from one independent variable to one dependent variable. From one amount of money to one specific food.

Ex #6: 1 9 4 4 -3 18 0 -6 -7 -1.5

This is a function since each value on the left corresponds to only ONE value on the right! Ex #7: 1 1 10 2 3 100 4

This is not a function because the value 1 is sent to TWO values, 1 & 2.

Graphing Functions

 One the graph above you can see that for every “x” value there is only ONE “y” value

 When graphing a relation you can tell if it is a FUNCTION by using what is known as the vertical line test

Definition 1.7: The Vertical Line Test (VLT) is a simple test to find out whether a graph is a function or not. To pass the VLT you must be able to draw a vertical line anywhere of the graph and have that line hit the relation only ONCE.

Ex. #8:  In the above diagram all the vertical lines only intersect the relation once. Therefore, this relation is a function

Ex. #9

 In this graph we have one vertical line that intersects our relation twice. Once one line fails the VLT the relation is not a function

Functions as Equations

 All graphs can be represented by equations. Like y= mx+ b y= x2 y = 3x2 + 4x + 5  “x” usually represents the independent variable  “y” usually represents the dependent variable

 To see if an equation is a function it must have only one value of “y” for each individual value of “x”

Grade 11 - Pg. 10 #1 - 5, 8, 11-14

 Review of Domain and Range notation

Grade 12 – Pg. 11 #1, 3, 4, 5, 7, 10 - 12