Introduction to Graphs
2 is a real number, and 3 is a real number. We can take those two numbers
and write them as a pair of real numbers: (2, 3). When we write a pair of
real numbers, the order is important. That is to say that (2, 3) is not the
same pair as (3, 2).
Unfortunately, (2, 3) is also the way we write the interval of real numbers
between 2 and 3. We have to try hard to never confuse a pair of numbers
for an interval, but it’s usually clear from the context of a problem whether
(2, 3) refers to a pair of numbers or to an interval.
R2 p
is the set of all pairs of real numbers. So (2, 3) 2 R2 , and (3, 2) 2 R2 ,
and ( 2, 7) 2 R2 , etc.. Any pair of real numbers is called a point in R2 .
Suppose f : A ! B is a function with A ✓ R and B ✓ R. The graph of f
is the subset of R2 consisting of all points of the form (a, f (a)).
Examples.
• If f : R ! R is the function f (x) = 5x, then f (3) = 5(3) = 15. We
put 3 in, and got 15 out. That means the point (3, 15) is in the graph of f .
Also, f (1) = 5, so (1, 5) is a point in the graph of f , and (2, f (2)) = (2, 10)
is a point in the graph of f as well.
• Suppose g : R ! R is the function where g(x) = x 2. If you put 2
in to g, then 0 comes out. That means the point (2, 0) is in the graph of g.
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56
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Drawing R22
Drawing
2a plane. The first coordinate of a point in R2 measures the
The set R2Ris
Drawing
R
2
2
The set RThe
is a plane.number
The first
coordinate
of a point in R 2measures the
horizontal.
measures
the vertical.
The set R2 issecond
a plane. The first
coordinate
of a point in R measures the
horizontal. The second number measures the vertical.
horizontal.
second
thecreates
vertical.
The set of The
points
in R2number
of the measures
form (x, 0)
a horizontal line called
2
The
set
of
points
in
R
of
the
form
(x,
0)
creates
a
horizontal line called
theThe
x-axis.
set of points in R2 of the form (x, 0) creates a horizontal line called
the x-axis.
the
Thex-axis.
set of points in R2 of the form (0, y) creates a vertical line called the
The set of points in R2 2of the form (0, y) creates a vertical line called the
y-axis.
The set of points in R of the form (0, y) creates a vertical line called the
y-axis.
y-axis.
2
57
46
Drawing
graphs
Drawing
graphs
Drawing
graphs
2
The
graph
of aisfunction
isofa R
subset
R2 . You
it byall
marking
The graph
of
a
function
a
subset
.
You
it bydraw
ofall of all of
2 ofdraw
The graph of a function is a subset of R . You draw
itmarking
by marking
the
points
in the graph.
Drawing
the points
in the
graph.
the points
ingraphs
the graph.
TheGraphs
graph of aoffunction
is a subset
of R2 . You draw it by marking all of
important
functions
Graphs
of important
functions
Graphs
ofthe
important
functions
the
points in
graph.
Some
functions
are
important
in mathematics
that you
be
SomeSome
functions
are important
enough
in enough
mathematics
that that
you should
be should
functions
are important
enough
in mathematics
you should
be
able their
to
their
graphs
quickly
you will be to
required to do so on
of draw
important
functions
able Graphs
to draw
graphs
quickly
(and
you (and
will
able
to draw
their
graphs
quickly
(and
you be
willrequired
be requireddotosodoonso on
A list
ofimportant
these important
functions
includesthat
constant
functions;
exams).
Aexams).
list
these
important
functions
includes
constant
functions;
the the
Some
functions
are
enough
in mathematics
you
should
be the
exams).
Aoflist
of these
important
functions
includes
constant
functions;
n
n
n f (x) = x for n ⇥ N odd
identity
function
id;
ffor
(x)
= even
x (and
forn an
even
n be
⇥=N;
identity
function
id;
f (x)
=(x)
xn =
⇥you
N;
f (x)
xrequired
for nn ⇥toNdo
oddso on
nan
able
to
draw
their
graphs
quickly
will
identity
function
id;
f
x
for
an
even
n
⇥
N;
f
(x)
1
1= x for n ⇥ N odd
1
1
and(x)
n =3ofn; fthese
(x)
=important
forN;odd
n f⇥(x)
N; =and
f (x) =
even n ⇥ N.the
n ⇥
and nexams).
for
and
⇥ N.
1 odd
xn
xnn for
functions
includes
constant
x =
xn for1 even
and 3n ; f A
3 ;list
f (x)
nfunctions;
⇥ N.
xn for oddn n ⇥ N; and f (x) = xn for even
n
identity function id; f (x) = x for an even n 2 N; f (x) = x for n 2 N odd
and n 3 ; f (x) = x1n for odd n 2 N; and f (x) = x1n for even n 2 N.
3
47
58
3
4859
53
4
4
Vertical line test
Vertical you’ll
line see
test
Sometimes
something drawn in R2 that looks like it might be the
Vertical
line
test
Sometimes you’ll see something drawn in R2 that looks like it might be the
graph of a function. To know for sure if it is, use
the vertical line test:
2
Sometimes
you’ll
see
something
drawn
in
R
that
looks
like itline
might
graph of a function. To know for sure if it is, use the
vertical
test:be the
graphIf of
a function.
know for
sure ifinit more
is, usethan
the one
vertical
line
test:
a vertical
line To
intersects
a thing
point,
then
a vertical
intersects a thing in more than one point, then
theIf thing
is notline
a graph.
Ifthat
a vertical
line
intersects
a thing
in more than one point, then
thing is not a graph of
a function.
the thing
a graph.
The reason
suchisa not
thing
is not a graph, is because if a vertical line intersects
such
a thing
is not
graphwould
of a function,
because
if apoints
vertical
it inThe
tworeason
different
points,
then
theathing
include is
two
different
The
reason
such
a
thing
is
not
a
graph,
is
because
if
a
vertical
line
intersects
linethe
intersects
it incoordinate
two di↵erent
then(1,the
include
two
with
same first
– forpoints,
example,
4) thing
and (1,would
9). This
could
itdi↵erent
in two different
points,
then
the
thing
would
include
two
different
points
same first
coordinate
– forthen
example,
(1, 4) and
(1, 9).
not be the points
graph with
of a the
function,
because
if it were,
the function
would
with
the
same
first
coordinate
–
for
example,
(1,
4)
and
(1,
9).
This
could
This two
could
not be the
graph–of4aand
function,
if it
were,ofthen
function
assign
different
numbers
9 – to because
the same
object
thethe
domain
–
not
be the
graph
of
a function,
because
ifand
it were,
then
the
function
would
would
assign
two
di↵erent
numbers
—
4
9
—
to
the
same
object
of
the
1. Functions can’t do that.
assign
two
numbers
4 and
9 – to the same object of the domain –
domain
—different
1. Functions
can’t– do
that.
1. Functions
that.the graph of a function. It fails the vertical line
Example.
A can’t
circle do
is not
Example. A circle is not the graph of a function. It fails the vertical line
test.
Example.
A circle is not the graph of a function. It fails the vertical line
test.
test.
49
60
5
Intercepts
Intercepts
Intercepts
If
the
graph
of
function
contains
point
of
the
form
(a,
0)
for
some
⇥
R,
If the
the graph
graph of
of aaa function
function contains
contains aaa point
point of
of the
the form
form (a,
(a,0)
0) for
for some
some aaa ⇥
2 R,
R,
If
then
is
called
an
x-intercept
of
the
graph.
then aaa is
is called
called an
an x-intercept
x-intercept of
of the
the graph.
graph.
then
If
the
graph
of
function
contains
point
of
the
form
(0,
b)
for
some
⇥
R,
If the
the graph
graph of
of aaa function
function contains
contains aaa point
point of
of the
the form
form (0,
(0,b)
b) for
for some
some bbb ⇥
2 R,
R,
If
then
b
is
called
the
y-intercept
of
the
graph.
then bb isis called
called the
the y-intercept
y-intercept of
of the
the graph.
graph.
then
Example.
Below
is
the
graph
of
function
The
x-intercepts
of
the
graph
Example. Below
Below is
is the
the graph
graph of
of aaa function
function fff... The
The x-intercepts
x-intercepts of
of the
the graph
graph
Example.
are
and
2.
The
y-intercept
of
the
graph
is
3.
are 222 and
and 2.
2. The
The y-intercept
y-intercept of
of the
the graph
graph is
is 3.
3.
are
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*
**
50
61
6
*
**
*
**
*
**
*
**
*
**
*
**
Little
circles
vs.
giant
dots
Little
circles
vs.
giant
dots
Little
circles
vs.
dots
Little
circles
vs.
giant
dots
Little
circles
vs.giant
giant
dots
Drawing
giant
dot
in
graph
means
that
point
ininthe
the
graph
ofofthe
the
Drawing
giant
dot
graph
means
that
point
in
the
graph
the
Drawing
aaaagiant
dot
inin
aaaagraph
means
that
point
isisisisin
graph
ofof
Drawing
giant
dot
inin
graph
means
that
point
the
graph
the
Drawing
a
giant
dot
a
graph
means
that
point
is
in
the
graph
of
the
function.
function.
function.
function.
function.
Drawing
little
circle
in
graph
means
that
point
not
inin
the
graph
ofof
Drawing
circle
aagraph
means
that
point
not
the
graph
Drawing
aaalittle
circle
inin
ain
means
that
point
isisisis
not
inin
the
graph
ofof
Drawing
aalittle
little
circle
in
agraph
graph
means
that
point
not
the
graph
Drawing
little
circle
a
graph
means
that
point
is
not
in
the
graph
of
the
function,
but
some
nearby
points
are.
the
function,
but
some
nearby
points
are.
the
function,
but
some
nearby
points
are.
the
function,
but
some
nearby
points
are.
the function, but some nearby points are.
Example.
Below
the
graph
ofof
the
function
: [1,
4)
⇥
where
Example.
Below
the
graph
the
function
4)
⇥
RRR
where
Example.
Below
isisisis
the
graph
ofof
the
function
fff:f:f[1,
4)
!
R
where
Example.
Below
the
graph
the
function
:[1,
[1,
4)
⇥
where
Example.
Below
is
the
graph
of
the
function
:
[1,
4)
⇥
R
where
f
(x)
=
x
+
2.
The
number
1
is
in
the
domain
of
f
,
and
f
(1)
=
3,
so
the
point
f
(x)
=
x
+
2.
The
number
1
is
in
the
domain
of
f
,
and
f
(1)
=
3,
so
the
point
f (x)
=
x
+
2.
The
number
1
is
in
the
domain
of
f
,
and
f
(1)
=
3,
so
the
point
ff(x)
=
x
+
2.
The
number
1
is
in
the
domain
of
f
,
and
f
(1)
=
3,
so
the
point
(x)
=
x
+
2.
The
number
1
is
in
the
domain
of
f
,
and
f
(1)
=
3,
so
the
point
(1,
3)
is
in
the
graph
of
f
.
We
can
label
it
with
a
giant
dot.
(1,
3)
is
in
the
graph
of
f
.
We
can
label
it
with
a
giant
dot.
(1,(1,
3)
is
in
the
graph
of
f
.
We
can
label
it
with
a
giant
dot.
3)
is
in
the
graph
of
f
.
We
can
label
it
with
a
giant
dot.
(1,
3)number
is in the44 graph
of
fthe
. We
can label
itbut
with
a giant
dot. really
The
number
not
in
domain
ofof
but
some
numbers
really
close
to
The
isisis
not
in
the
domain
of
ff,f,,but
some
numbers
close
to
The
number
4
is
not
in
the
domain
of
f
some
numbers
really
close
to
The
number
4
not
in
the
domain
,
but
some
numbers
really
close
toto
The
number
4
is
not
in
the
domain
of
f
,
but
some
numbers
really
close
4
are.
If
4
was
in
the
domain,
then
f
(4)
=
6,
and
(4,
6)
would
be
a
point
in
4
are.
If
4
was
in
the
domain,
then
f
(4)
=
6,
and
(4,
6)
would
be
a
point
in
4 are.
If If
4Ifwas
in in
the
domain,
then
f (4)
==
6,
and
(4,(4,
6)6)
would
bebe
a point
in inin
44 are.
44was
domain,
then
ff(4)
and
would
aapoint
are.
was
inthe
the
domain,
then
(4)
=6,
6,
and
(4,
6)
would
be
point
the
graph
of
f
.
But
4
isn’t
in
the
domain,
so
(4,
6)
isn’t
a
point
in
the
graph
the
graph
of
f
.
But
4
isn’t
in
the
domain,
so
(4,
6)
isn’t
a
point
in
the
graph
the
graph
of of
fof.ffBut
4 isn’t
in in
the
domain,
soso
(4,
6)6)
isn’t
a point
in in
the
graph
the
graph
. . But
44isn’t
the
domain,
(4,
isn’t
aapoint
the
graph
the
graph
But
isn’t
in
the
domain,
so
(4,
6)
isn’t
point
in
the
graph
of
f
.
The
graph
does
go
all
the
way
up
to
the
point
(4,
6),
but
it
doesn’t
of
f
.
The
graph
does
go
all
the
way
up
to
the
point
(4,
6),
but
it
doesn’t
of ofof
f .ff.The
graph
does
go
all
the
way
up
to
the
point
(4,
6),
but
it
doesn’t
The
graph
does
go
all
the
way
up
to
the
point
(4,
6),
but
it
doesn’t
. the
The
graph(4,
does
go
all
the
way
uppoint
to the
point
(4,a6),
but circle
it doesn’t
include
the
point
(4,
6).
So
we
label
the
point
(4,
6)
with
aalittle
little
circle
to
include
point
6).
So
we
label
the
(4,
6)
with
little
to
include
the
point
(4,
6).
So
we
label
the
point
(4,
6)
with
a
circle
to
include
the
point
(4,
6).
So
we
label
the
point
(4,
6)
with
little
circle
toto
include
the
point
(4,
6).
So
we
label
the
point
(4,
6)
with
a
little
circle
remind
us
that
it’s
not
actually
in
the
graph.
remind
us
that
it’s
not
actually
in
the
graph.
remind
usus
that
it’sit’s
not
actually
in in
the
graph.
remind
that
not
actually
graph.
remind
us
that
it’s
not
actually
inthe
the
graph.
Example.
Below
the
graph
ofof
the
function
:R
{2}
⇥
where
Example.
Below
the
graph
the
function
{2}
⇥
where
Example.
Below
isisisis
the
graph
ofof
the
function
ggg:g:R
!
RRRR
where
Example.
Below
the
graph
the
function
:R
RR{2}
{2}
⇥
where
Example.
Below
is
the
graph
of
the
function
g
:
{2}
⇥
R
where
g(x)
=
4.
Since
2
is
not
in
the
domain
of
g,
the
point
(2,
4)
is
not
inin
the
g(x)
=
4.
Since
2
is
not
in
the
domain
of
g,
the
point
(2,
4)
is
not
the
g(x)
= =4. 4.Since
2 2is isnot
in inthethedomain
of ofg, g,thethepoint
(2,(2,
4) 4)
is is
not
inin
the
g(x)
Since
not
domain
point
not
the
g(x)
=
4.
Since
2
is
not
in
the
domain
of
g,
the
point
(2,
4)
is
not
in
the
graph
of
g,
so
we
label
it
with
a
little
circle
to
remind
us
that
it’s
not
in
the
graph
of
g,
so
we
label
it
with
a
little
circle
to
remind
us
that
it’s
not
in
the
graph
of
g,
so
we
label
it
with
a
little
circle
to
remind
us
that
it’s
not
in
the
graph
of
g,
so
we
label
it
with
a
little
circle
to
remind
us
that
it’s
not
in
the
graph of g, so we label it with a little circle to remind us that it’s not in the
graph.
graph.
graph.
graph.
graph.
6251751
7
Domainsand
and
Ranges
for
graphs
Domains
Ranges
for
graphs
Domains
and
Ranges
for
graphs
Suppose you
are given
a graph,
and you’re
told that
it is the
graph of
Suppose
Supposeyou
youare
aregiven
givena agraph,
graph,and
andyou’re
you’retold
toldthat
thatit itis isthe
thegraph
graphofof
a function.ToTo
find
the
domain
the
function,
draw
its
“shadow”
on
the
a afunction.
find
the
domain
ofofof
the
function,
draw
itsits
“shadow”
onon
the
function.
To
find
the
domain
the
function,
draw
“shadow”
the
x-axis.
x-axis.
x-axis.
To
find the
range of the
function, draw
its “shadow”
on the
y-axis.
ToTo
find
findthe
therange
rangeofofthe
thefunction,
function,draw
drawitsits“shadow”
“shadow”ononthe
they-axis.
y-axis.
Example.
Drawn
below
is the
graph
the
function
. The
domain
Example.
Drawn
below
is isthe
graph
ofofof
the
function
f .ffThe
domain
ofoffoffisf isis
Example.
Drawn
below
the
graph
the
function
.
The
domain
the
set
real
numbers
the
x-axis
that
lie
directly
below
the
graph.
Those
the
setset
ofofof
real
numbers
ininin
the
x-axis
that
lielie
directly
below
the
graph.
Those
the
real
numbers
the
x-axis
that
directly
below
the
graph.
Those
are
all
the
numbers
between
2 and
Because
there
a giant
dot
on
the
are
allall
ofofof
the
numbers
between
2 2and
5.5.5.
Because
there
is isaisagiant
dot
onon
the
are
the
numbers
between
and
Because
there
giant
dot
the
2
2R , we know that 5 is in the domain. But since there is a little
point
(5,
3)
point
(5,(5,
3)3)
2 RR
, 2we
that
5 is in the domain. But since there is a little
point
, weknow
know
2 5 is in the domain. But since there is a little
2that
circle
on
the
point
(2,
1)
R
,
know
that
is not
domain.
That
circle on the point (2, 1) 2 R , 2wewe
know
that
2 is2 not
in in
thethe
domain.
That
is, is,
circle
on
the
point
(2,
1)
R
,
we
know
that
2
is
not
in
the
domain.
That
is,
domain
f the
is the
interval
thethe
domain
of of
f is
interval
(2,(2,
5]. 5].
theThe
domain
of f fis isthe
interval
(2, 5].
range
numbers
y-axis
that
directly
The range
of of
f is thethe
setset
of of
realreal
numbers
on on
thethe
y-axis
that
lie lie
directly
to to
The
range
of
f
is
the
set
of
real
numbers
on
the
y-axis
that
lie
directly
to
graph
. The
range
is the
interval
thethe
leftleft
of of
thethe
graph
of of
f . fThe
range
of of
f isf the
interval
(1,(1,
5]. 5].
the left of the graph of f . The range of f is the interval (1, 5].
* * * * * * * * * * * * * * * * * * * * * * * ** *
*
*
*
*
*
*
*
*
*
*
*
*
*
63 8
52
Exercises
~‘tS~‘tS
-z
i -z i
Exercises
For #1-4, decide whether Exercises
or not
each of the drawings in R2 is the graph of
Exercises
aFor
function.
#1-4, decide whether or not each of the drawings in R2 is the graph of
For #1-4,
decide
whether
or each
not each
of drawings
the drawings
the graph
#1-4,
decide
whether
or not
of the
in R2inisR2theis graph
of of
aFor
function.
a function.
a function.
2 is the graph of
For#1-4,
#1-4,decide
decidewhether
whetherorornot
noteach
eachofofthe
thedrawings
drawingsininRJR2
For
is the graph of
function.
a afunction.
Exercises
Exercises
00
00
#3) #3)
4tq.)4tq.)
UU
For
For#5-6,
#5-6,list
listthe
thex-z and
and y-intercepts
y-intercepts of
of the
the graphs
graphs below.
below.
3
2
53
—2
-5-4 4-z 4
58
53
53
-3
6458
~i.4
For #5-8, determine the domains and ranges for the functions that are
drawn.
For #5-8, determine the domains and ranges for the functions that are
drawn.
For #7-12, determine the domains and ranges for the functions that are
drawn.
*7)
(s, q)
*7)
(s, q)
(‘i~,z)
(‘i~,z)
(I-I)
(a,’)
(I-I)
(a,’)
49)
49)
(L,-I)
(-2,-5)
(L,-I)
*i*i
i)(-2,-5)
i)
#12)
#12)
aa
S S
59
59
65
-a-a
13.) Write out the product (x + y)4 .
For #14-19, assume that f (x) = x + 1, g(x) = 4, and that h(x) = x2 2.
Match each of the numbered functions with one of the lettered formulas.
14.) f
g(x)
17.) h g(x)
A.) x2
D.) 5
1
15.) g f (x)
16.) g h(x)
18.) f
19.) h f (x)
h(x)
B.) x2 + 2x
1
C.) 4
E.) 14
20.) What is the implied domain of f (x) = 2x2
21.) What is the implied domain of g(x) =
66
2x
x 8
3x + 7 ?
?