MCR3U UNIT 1 – INTRODUCTION TO FUNCTIONS Textbook: Nelson: Functions
Date Lesson Text TOPIC Homework
Feb. 4 1.1
Graphing Relations Look over website, complete
the info sheet and get
permission to post signed.
Bring in $2 for lesson shells &
$7 if you need a calculator.
WS 1.1
Answers: 1. (4,2) 2. (2,-
1) 3. (2,5), (-1,2) 4. (-1,-8),
(3, -8) 5. (0,1),
(1, -2) 6. (4,2)
Feb. 5 1.2 1.1 Functions Pg. 10 #1, 2, 6, 7 (sketch, not
graph), 8,9, 11, 12, 15
Feb. 6 1.3 1.2 Function Notation Pg. 22 # 1 - 3, 5 - 8, 10, 11, 12,
13, 17
Feb. 7 1.4 1.4 Domain and Range Pg. 35 #1 - 7, 9a-e, 10ab, 11,
14
Feb. 8 1.5 1.5 The Inverse Function
Q1 (1.1 – 1.3)
Pg. 46 # 1, 3 – 5, 8, 9a-e,
10 – 13, 16, 20
Feb.
11 1.6
1.6
-1.8
Transformations WS 1.6
Feb.
12 1.7
1.6
-1.8
Transformations
Horizontal Stretches
WS 1.7
Feb.
13 1.8
1.6
-1.8
Transforming Radical and Reciprocal
Functions
WS 1.8
Feb.
14 1.9
Review for Unit 1 Test Pg. 76 # 1 – 6, 7ab, 8, 11, 12,
13a, 15, 17, 19
Pg. 78 # 1, 2, 3ab, 5, 7
Feb.
15 1.10
UNIT 1 TEST
WEBSITE: mrkennedy.pbworks.com
1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
y
1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x
1
2
3
4
5
6
7
8
–1
–2
–3
–4
–5
–6
–7
–8
y
2 4 6 8 10–2–4–6–8–10 x
2
4
6
8
10
–2
–4
–6
–8
–10
y
MCR 3U Lesson 1.1 Graphing Relations
1. Graph each of the following relations:
a) 12 xy and xy 5
b) 01223 yx and 036 yx
c) 2xy , 34
2 xy , and 8)7(2 2 xy
WS 1.1
MCR 3U Lesson 1.2 Functions
1. Consider the data which lists the 1) favourite hockey team and 2) number of siblings.
When the data is presented in this way, we consider “Favourite Team” to be the
________________________ variable and “# of Siblings” to be the
________________________ variable.
For this data we can create a corresponding mapping diagram .
We can also list the Domain _______________________________________________
and the Range ________________________ of this relation.
2. Consider the data at right. When the data is presented in this way, we consider “#
of Siblings” to be the ________________________ variable and “Favourite
Team” to be the ________________________ variable.
For this data we can create a corresponding mapping diagram .
We can also list the Domain _______________________________________________
and the Range _______________________________________ of this relation.
The relation in example 2 is: ___________________________________
Favourite
Team
# of
Siblings
Ottawa 2
Pittsburgh 3
Toronto 0
Montreal 3
Boston 1
# of Siblings
Favourite
Team
2 Ottawa
3 Pittsburgh
0 Toronto
3 Montreal
1 Boston
The relation in example 1 (above) is called a _____________________ because:
___________________________________________________________________
___________________________________________________________________
2 4 6 x
–2
–4
–6
y
6 8 10 12 x
6
8
10
12
y
1 2 3 4 5–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
3. Classify each of the following as a function (F) or not (N):
a) PhilEngJillFrBillEng .,,.,,., b)
c)
4. Plot the relations listed above in 3c) and 3d.
5. Classify each of the following as a function (F) or not (N):
a) b) 23 xy c) 2xy d)
2yx
Ottawa Senators
Number Name
5 Ceci
9 Ryan
65 Karlsson
41 Anderson
x y
1 -3
2 -1
3 0
4 -1
Shoe size
# of friends
9 8
8.5 12
9.5 10
9 7
7 13
d)
One way of determining if a
graphed relation is a function is
to use the Vertical Line Test.
Pg. 10 #1, 2, 6, 7 (sketch, not graph), 8,9, 11, 12, 15
1 2 3 4 5 6 x
1
2
3
4
5
y
2 4 6 8 x
2
4
6
–2
–4
–6
y
2 4 6 x
2
4
6
y
1 2 3 4 5 6–1–2–3–4–5–6 x
1
2
3
4
5
6
–1
–2
–3
–4
–5
–6
y
MCR 3U Lesson 1.3 Function Notation
Function Notation: Notation used to represent the value of the dependent variable (the output) for a given value
of the independent variable (the input).
Symbols such as f(x) and g(x) are used to represent the dependent variable y for a given value of the independent
variable x. Foe this reason, y and f(x) are interchangeable in the equation of a function, so y = f(x).
Function f. Function g.
Ex. 1 Consider the function 32)( xxf .
a) Graph this function. b) Determine each value:
i) )1(f . ii) 122 f
iii) xf 54 iv) 3
6
f
Ex. 2 The bottom of an inclined conveyor belt is 1.5 metres above the ground. A toy placed on this belt rises at a
rate of 0.5 m/sec.
a) Use function notation to write an equation for this
situation.
b) What will be the height of the toy after 4 seconds?
c) How long will it take for the toy to reach a height of 6
metres?
Ex. 3 Given f(x) = 2x – 3 and g(x) = x2 – 4x + 5, find the value(s) of a for which f(a) = g(a).
Ex. 4 Explain function notation and compare it with other ways to write equations. Include an example.
p. 22 #1acf, 2, 5ac, 6, 7, 8, 10,
11ac, 12, 13, 17
1 2 3 4 5 6–1–2–3–4–5–6 x
1
2
3
4
5
6
–1
–2
–3
–4
–5
–6
y
MCR 3U Lesson 1.4 Domain and Range
Ex. 1 State the domain and range of each relation:
Domain: __________________
Range: ___________________
Domain: ________________
Range: _________________
Horizontal Asymptote
y = 0 -tells us how the curve
behaves at each end of the
x-axis.
Asymptotes are broken lines.
Domain: __________________
Range: ___________________
Domain: ______________
Range: ________________
Ex. 2 State the domain and range of
a) 4352 xy .
b) xy
c) x
y10
d) ,52 xy 7x
1 2 3 4 5 6–1–2–3–4–5–6 x
1
2
3
4
5
6
–1
–2
–3
–4
–5
–6
y
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
Vertical Asymptote – tells us where the
curve is undefined x = 0
Pg. 35 # 1 – 7, 9a-e, 10ab, 11, 14
MCR 3U Lesson 1.5 Inverse of a Function
The inverse of a function is the reverse of the original function. It undoes what the original has done and
can be found using the inverse operations of the original function in reverse order.
The inverse of a function is not necessarily a function.
The symbol )(1 xf is the notation for the inverse function of f .
Ex. 1 The cost to rent a hall for a party is $200 plus $10 per guest.
a) Use function notation to write an equation b) What would be the cost for 80 guests to attend?
representing this relation, where the cost of
the party is the dependent variable.
c) Use function notation to write an equation d) How many guests could attend for $960?
representing this relation, where the number
of guests is the dependent variable.
Ex. 2 For each of the following functions, find the equation of the inverse function for i) by reversing
the operations, AND for ii) by interchanging the variables
i) 15)( xxg
ii) xxh4
13)(
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
2 4 6 x
2
4
6
y Ex. 3
Ex. 3
a) Plot )4,6(),2,5(),1,3(),4,2(f
b) List 1f _________________________ .
c) Plot 1f on the same axes.
d) What do you notice about 1f ?
Ex. 4 Given that 32 xxg ,
a) Plot g and 1g on the same axes.
b) State the defining equation for 1g .
c) Evaluate: 51 1 gg
d) Evaluate:
2
2102
1
g
g
Pg . 46 #3 - 5, 8abc, 9a-e, 10 ,12,13, 16,17, 20
The domain of f is the range of 1( )f x and the range of f is the domain of 1( )f x .
1( )f x is the reflection of f in the line y x .
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
MCR 3U Lesson 1.6 Transformations (Part 1)
Ex. 1
a) If 2xxf , what is 3xf ?
b) Plot xf and 3xf on the axes at right.
Ex. 2
a) If xxg , what is 6xg ?
b) Plot xg and 6xg on the axes at right.
Given the graph of xf , the graph of axf will be a
Given the graph of xf , the graph of axf will be a
Ex. 3 Given the graph of xf at right, plot each of the following on
the same axes:
a) 3xf
b) 2xf
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
Ex. 4
a) If 2xxf , what is 4xf ?
b) Plot xf and 4xf on the axes at right.
Ex. 5
c) If xxg , what is 3xg ?
d) Plot xg and 3xg on the axes at right.
Given the graph of xf , the graph of axf will be a
Given the graph of xf , the graph of axf will be a
Ex. 6 Given the graph of xf at right, plot each of the following on
the same axes:
a) 3xf
b) 4xf
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
y
Ex. 7
c) If 2xxf , what is xf ?
d) Plot xf and xf on the axes at right.
Ex. 8
e) If xxg , what is xg2 ?
f) Plot xg and xg2 on the axes at right.
Given the graph of xf , the graph of xf will be a
Given the graph of xf , the graph of xaf will be a
Ex. 9 Given the graph of xf at right, plot each of the following on
the same axes:
a) xf
b) xf2
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
Ex. 10 Given the graph of xf at right, plot each of the
following on the same axes:
a) 63 xf
b) xf2
1
Ex. 11 Given the graph of xf at right, plot each of the
following on the same axes:
a) 4 xf
b) 12 xf
Ex. 12 Given the graph of xf at right, plot each of the
following on the same axes:
a) 512 xf
b) 23 xf
WS 1.6
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
–1
–2
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
MCR 3U Lesson 1.7 Transformations (Part 2) - Horizontal Stretches
At right is the graph of 2)( xxf .
If 2)( xxf , then determine:
)2( xf
Now graph y ___________ .
This, in this case, is the graph of )2( xf .
So, the transformation )2( xf can be described as a
_____________________________________
_____________________________________
Ex. 1 Given the graph of xf , plot each of the following:
a) xf 2 b)
xf
2
1
1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
–1
–2
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
At right is the graph of 2)( xxf .
If 2)( xxf , then determine:
)62( xf
Now use a table of values to graph
y ______________________ .
So, the transformation )62( xf can be described as a
____________________________________________________________
____________________________________________________________
Ex. 2 Given the graph of xf , plot each of the following:
a) 42 xf b) xf
Ex. 3 Given the graph of xf at right, plot each
of the following on the same axes:
a) 3 xf
b) 52 xf
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
Ex. 4 Given the graph of xf at right, plot each
of the following on the same axes:
a) xf
b) 622 xf
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
Ex. 5 Given the graph of xf at right, plot each
of the following on the same axes:
a) 51 xf
b)
xf
2
13
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
WS 1.7
2 4 6 8 10–2–4–6–8–10 x
2
4
6
8
10
–2
–4
–6
–8
–10
y
2 4 6 8 10–2–4–6–8–10 x
2
4
6
8
10
–2
–4
–6
–8
–10
y
MCR 3U Lesson 1.8 Graphing Radical and Reciprocal Functions
Ex. 1 Graph 43x2y
Ex. 2 Graph 1x2
1y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
Ex. 3 x
xf1
. Graph xf , and xf4 on the grids provided below:
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
2 4 6 8–2–4–6–8 x
2
4
6
8
–2
–4
–6
–8
y
Ex. 4 Graph 12x
6y
.
WS 1.8
