1.1
CHECK Your Understanding
1. State the domain and range of each relation. Then determine
whether the relation is a function, and justify your answer.
a)
y
4
c)
2
–12 –8 –4
0
–2
x
4
d) y 5 3x 2 5
8
e)
–4
–6
b)
4
y
2
–4 –2 0
–2
5 (1, 4), (1, 9), (2, 7),
(3, 25), (4, 11)6
–4
0
–3
1
1
2
2
3
x
2
4
6
f ) y 5 25x 2
–4
–6
2. State the domain and range of each relation. Then determine whether
the relation is a function, and justify your answer.
a) y 5 22(x 1 1) 2 2 3
b) y 5
1
x13
c) y 5 22x
e) x 2 1 y 2 5 9
d) y 5 cos x 1 1
f ) y 5 2 sin x
PRACTISING
3. Determine whether each relation is a function, and state its domain
and range.
a)
c)
1
3
5
7
2
4
6
b) 5(2, 3), (1, 3), (5, 6),
(0, 21)6
NEL
e)
0
1
2
3
2
1
4
10
100
d) 5(2, 5), (6, 1), (2, 7),
(8, 3)6
0
1
2
3
f ) 5 (1, 2), (2, 1), (3, 4),
(4, 3)6
Chapter 1
11
4. Determine whether each relation is a function, and state its domain
K
a)
8
and range.
y
b)
6
6
4
4
2
2
0
–2
x
–4 –2 0
–2
2
4
6
c) x 2 5 2y 1 1
y
d) x 5 y 2
x
4
8
12
16 20
3
x
f ) f (x) 5 3x 1 1
e) y 5
–4
5. Determine the equations that describe the following function rules:
a) The input is 3 less than the output.
b) The output is 5 less than the input multiplied by 2.
c) Subtract 2 from the input and then multiply by 3 to find the output.
d) The sum of the input and output is 5.
6. Martin wants to build an additional closet in a corner of his bedroom.
l
Because the closet will be in a corner, only two new walls need to be
built. The total length of the two new walls must be 12 m. Martin wants
the length of the closet to be twice as long as the width, as shown in the
diagram.
a) Explain why l 5 2w.
b) Let the function f (l ) be the sum of the length and the width. Find
the equation for f (l ).
c) Graph y 5 f (l ).
d) Find the desired length and width.
wall
w
closet
wall
7. The following table gives Tina’s height above the ground while riding a
A
Ferris wheel, in relation to the time she was riding it.
Time (s)
0
20
40
60
80
100
120
140
160
180
200
220
240
Height (m)
5
10
5
0
5
10
5
0
5
10
5
0
5
a) Draw a graph of the relation, using time as the independent variable
and height as the dependent variable.
What is the domain?
What is the range?
Is this relation a function? Justify your answer.
Another student sketched a graph, but used height as the independent
variable. What does this graph look like?
f ) Is the relation in part e) a function? Justify your answer.
b)
c)
d)
e)
12
1.1
Functions
NEL
1.1
8. Consider what happens to a relation when the coordinates of all its
ordered pairs are switched.
a) Give an example of a function that is still a function when its
coordinates are switched.
b) Give an example of a function that is no longer a function when
its coordinates are switched.
c) Give an example of a relation that is not a function, but becomes a
function when its coordinates are switched.
9. Explain why a relation that fails the vertical line test is not a function.
10. Consider the relation between x and y that consists of all points (x, y)
such that the distance from (x, y) to the origin is 5.
a) Is (4, 3) in the relation? Explain.
b) Is (1, 5) in the relation? Explain.
c) Is the relation a function? Explain.
11. The table below lists all the ordered pairs that belong to the
function g(x).
x
0
1
2
3
4
5
g(x)
3
4
7
12
19
28
a) Determine an equation for g(x).
b) Does g(3) 2 g(2) 5 g(3 2 2) ? Explain.
12. The factors of 4 are 1, 2, and 4. The sum of the factors is
T
1 1 2 1 4 5 7. The sum of the factors is called the sigma function.
Therefore, f (4) 5 7.
a) Find f (6), f (7), and f (8).
c) Is f (12) 5 f (3) 3 f (4) ?
b) Is f (15) 5 f (3) 3 f (5) ?
d) Are there others that will work?
13. Make a concept map to show what you have learned about functions.
C
Put “FUNCTION” in the centre of your concept map, and include
the following words:
algebraic model
dependent variable
domain
function notation
graphical model
independent variable
mapping model
numerical model
range
vertical line test
Extending
14. Consider the relations x 2 1 y 2 5 25 and y 5 "25 2 x 2. Draw
the graphs of these relations, and determine whether each relation is
a function. State the domain and range of each relation.
15. You already know that y is a function of x if and only if the graph
Communication
Tip
A concept map is a type of
web diagram used for
exploring knowledge and
gathering and sharing
information. A concept map
consists of cells that contain a
concept, item, or question and
links. The links are labelled and
denote direction with an arrow
symbol. The labelled links
explain the relationship
between the cells. The arrow
describes the direction of the
relationship and reads like a
sentence.
passes the vertical line test. When is x a function of y? Explain.
NEL
Chapter 1
13
Answers
Chapter 1
Getting Started, p. 2
51
16
26
d) a 2 1 5a
(x 1 y) (x 1 y)
(5x 2 1) (x 2 3)
(x 1 y 1 8) (x 1 y 2 8)
(a 1 b) (x 2 y)
horizontal translation 3 units to the right,
vertical translation 2 units up;
c) 2
1. a) 6
2.
3.
b)
a)
b)
c)
d)
a)
y
10
8
6
4
2
x
–6 –4 –2
0
–2
2
4
6
b) horizontal translation 1 unit to the right,
vertical translation 2 units up;
y
10
8
6
4
2
x
–6 –4 –2
0
–2
2
4
6
c) horizontal stretch by a factor of 2, vertical
stretch by a factor of 2, reflection across
the x-axis;
4
2
x
–90° 0
–2
90°
270°
–4
–6
d) horizontal compression by a factor
1
of 2 , vertical stretch by a factor of 2,
reflection across the x-axis;
y
6
4
2
x
–2
0
–2
2
4
–4
–6
612
Answers
6
2.
3.
Lesson 1.1, pp. 11–13
y
6
–270°
4. a) D 5 5xPR 0 22 # x # 26,
R 5 5yPR 0 0 # y # 26
b) D 5 5xPR6, R 5 5 yPR 0 y $ 2196
c) D 5 5xPR 0 x 2 06,
R 5 5 yPR 0 y 2 06
d) D 5 5xPR6,
R 5 5 yPR 0 23 # y $ 36
e) D 5 5xPR6, R 5 5 yPR 0 y . 06
5. a) This is not a function; it does not pass
the vertical line test.
b) This is a function; for each x-value, there
is exactly one corresponding y-value.
c) This is not a function; for each x-value
greater than 0, there are two
corresponding y-values.
d) This is a function; for each x-value, there
is exactly one corresponding y-value.
e) This is a function; for each x-value, there
is exactly one corresponding y-value.
6. a) 8
b) about 2.71
7. If a relation is represented by a set of ordered
pairs, a table, or an arrow diagram, one can
determine if the relation is a function by
checking that each value of the independent
variable is paired with no more than one
value of the dependent variable. If a relation
is represented using a graph or scatter plot,
the vertical line test can be used to determine
if the relation is a function. A relation may
also be represented by a description/rule or
by using function notation or an equation.
In these cases, one can use reasoning to
determine if there is more than one value of
the dependent variable paired with any value
of the independent variable.
8
10
1. a) D 5 5xPR6;
R 5 5yPR 0 24 # y # 226; This is a
function because it passes the vertical
line test.
b) D 5 5xPR 0 21 # x # 76;
R 5 5 yPR 0 23 # y # 16; This is a
function because it passes the vertical
line test.
c) D 5 51, 2, 3, 46;
R 5 525, 4, 7, 9, 116; This is not a
function because 1 is sent to more than
one element in the range.
d) D 5 5xPR6; R 5 5yPR6; This is a
function because every element in the
domain produces exactly one element
in the range.
e) D 5 524, 23, 1, 26; R 5 50, 1, 2, 36;
This is a function because every element
of the domain is sent to exactly one
element in the range.
4.
5.
f ) D 5 5xPR6; R 5 5yPR 0 y # 06;
This is a function because every
element in the domain produces
exactly one element in the range.
a) D 5 5xPR6; R 5 5 yPR 0 y # 236;
This is a function because every
element in the domain produces exactly
one element in the range.
b) D 5 5xPR 0 x 2 236;
R 5 5 yPR 0 y 2 06; This is a function
because every element in the domain
produces exactly one element in the
range.
c) D 5 5xPR6; R 5 5 yPR 0 y . 06;
This is a function because every
element in the domain produces exactly
one element in the range.
d) D 5 5xPR6;
R 5 5 yPR 0 0 # y # 26; This is a
function because every element in the
domain produces exactly one element
in the range.
e) D 5 5xPR 0 23 # x # 36;
R 5 5yPR 0 23 # y # 36; This is not
a function because (0, 3) and (0, 3) are
both in the relation.
f ) D 5 5xPR6;
R 5 5yPR 0 22 # y # 26; This is a
function because every element in the
domain produces exactly one element
in the range.
a) function; D 5 51, 3, 5, 76;
R 5 52, 4, 66
b) function; D 5 50, 1, 2, 56;
R 5 521, 3, 66
c) function; D 5 50, 1, 2, 36; R 5 52, 46
d) not a function; D 5 52, 6, 86;
R 5 51, 3, 5, 76
e) not a function; D 5 51, 10, 1006;
R 5 50, 1, 2, 36
f ) function; D 5 51, 2, 3, 46;
R 5 51, 2, 3, 46
a) function; D 5 5xPR6;
R 5 5 yPR 0 y $ 26.
b) not a function; D 5 5xPR 0 x $ 26;
R 5 5 yPR6
c) function; D 5 5xPR6;
R 5 5 yPR 0 y $ 20.56
d) not a function; D 5 5xPR 0 x $ 06;
R 5 5 yPR6
e) function; D 5 5xPR 0 x 2 06;
R 5 5 yPR 0 y 2 06
f ) function; D 5 5xPR6; R 5 5 yPR6
a) y 5 x 1 3
c) y 5 3(x 2 2)
b) y 5 2x 2 5
d) y 5 2x 1 5
NEL
6. a) The length is twice the width.
3
b) f (l ) 5 l
2
c)
f(B)
12.
8
6
4
13.
2
f (6) 5 12; f (7) 5 8; f (8) 5 15
Yes, f (15) 5 f (3) 3 f (5)
Yes, f (12) 5 f (3) 3 f (4)
Yes, there are others that will work.
f (a) 3 f (b) 5 f (a 3 b) whenever a
and b have no common factors other
than 1.
Answers may vary. For example:
c) The absolute value of a number is always
greater than or equal to 0. There are no
solutions to this inequality.
d)
a)
b)
c)
d)
–10 –8 –6 –4 –2
5. a) 0 x 0 # 3
b) 0 x 0 . 2
6.
y
2
4
6
8
10
Height (m)
6
4
function
notation
2
vertical line
test
14.
12
14
16
y
x
2
4
6
8
a) The graphs are the same.
b) Answers may vary. For example,
x 2 8 5 2 (2x 1 8), so they are
negatives of each other and have the
same absolute value.
7. a)
4
2
x
–4 –2
250
Time (s)
0
graphical
model
dependent
variable
b) D 5 50, 20, 40, 60, 80, 100, 120,
140, 160, 180, 200, 220, 2406
c) R 5 50, 5, 106
d) It is a function because it passes the
vertical line test.
e)
y
10
10
2
algebraic
model
range
x
8
4
FUNCTION
8
50 100 150 200 250
Time (s)
6
6
mapping
model
10
0
4
domain
d) length 5 8 m; width 5 4 m
7. a)
y
200
0
–2
2
4
–4
150
100
6
50
0
2
4 6 8
Height (m)
y
b)
4
x
10
2
x
–4 –2
0
2
4
The first is not a function because it fails the
vertical line test:
D 5 5xPR 0 25 # x # 56;
R 5 5yPR 0 25 # y # 56.
The second is a function because it passes
the vertical line test:
D 5 5xPR 0 25 # x # 56;
R 5 5yPR 0 0 # y # 56.
15. x is a function of y if the graph passes the
horizontal line test. This occurs when any
horizontal line hits the graph at most once.
c)
Answers
f ) It is not a function because (5, 0) and
(5, 40) are both in the relation.
8. a) 5(1, 2), (3, 4), (5, 6)6
b) 5(1, 2), (3, 2), (5, 6)6
c) 5(2, 1), (2, 3), (5, 6)6
9. If a vertical line passes through a function
and hits two points, those two points have
identical x-coordinates and different
y-coordinates. This means that one
x-coordinate is sent to two different
elements in the range, violating the
definition of function.
10. a) Yes, because the distance from (4, 3) to
(0, 0) is 5.
b) No, because the distance from (1, 5) to
(0, 0) is not 5.
c) No, because (4, 3) and (4, 23) are
both in the relation.
11. a) g(x) 5 x 2 1 3
b) g(3) 2 g(2) 5 12 2 7
55
g(3 2 2) 5 g(1)
54
So, g(3) 2 g(2) 2 g(3 2 2).
NEL
numerical
model
independent
variable
2
8
B
–2 0
–2
0
c) 0 x 0 $ 2
d) 0 x 0 , 4
d)
Lesson 1.2, p. 16
1. 0 25 0 , 0 12 0, 0 215 0 , 0 20 0 , 0 225 0
2. a) 22
c) 18
e) 22
b) 235
d) 11
f ) 22
3. a) 0 x 0 . 3
c) 0 x 0 $ 1
b) 0 x 0 # 8
d) 0 x 0 2 5
4. a)
–10 –8 –6 –4 –2
b)
–20–16 –12 –8 –4
0
0
2
4
4
8
6
8
10
12 16 20
8.
When the number you are adding or
subtracting is inside the absolute value signs,
it moves the function to the left (when
adding) or to the right (when subtracting)
of the origin. When the number you are
Answers
613