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
4
5
10
50
15
20
25
50
250
**Define
Revenue**
 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:
 Is this a relation?
Shoe Size
Height (cm)
7.5
10
11.5
8.5
161
158
175
177
 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
4
-3
0
-7
9
4
18
-6
-1.5
This is a function since each value on the left
corresponds to only ONE value on the right!
Ex #7:
1
10
100
1
2
3
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  3x 2  4 x  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