2.1 - MATH
1081
 Wednesday,
January
19
 
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Unformatted text preview: MATH
1081
 Wednesday,
January
19
 
 Section
2.1
 
 PROPERTIES
OF
FUNCTIONS
 
 Homework
#1
(due
1/24):

 Section
1.1
#36,
64,
70,
72
 
 
 
 
 
 
 
 
 Section
2.1
#36,
44,
46,
64
 
 Starting
today,
clicker
questions
will
be
asked
in
each
lecture
 session.

Remember
that
it
is
½
point
for
an
incorrect
answer
 and
1
point
for
a
correct
answer,

also
½
point
for
check‐in
and
 ½
point
for
check‐out.
 
 To
ready
your
clicker,
press
and
hold
the
On/Off
button
until
 the
Power
light
flashes.

Then
enter
the
2‐key
frequency
code
 (one
letter
at
a
time)
as
indicated
on
the
sign
posted
at
the
 front
of
the
room.

At
that
point
you
should
see
the
Vote
Status
 light
flash
green.
 
 Clicker
Check­in:

Press
“A”
now
to
check
in.
 The
Vote
Status
light
should
again
flash
green
to
indicate
that
 your
vote
has
been
received. As
we
discussed
briefly
during
the
last
lecture,
for
much
of
this
 class,
we
will
be
interested
in
determining
the
rate
of
change
of
 a
graph
at
a
certain
point.

To
do
this
formally,
we
use
the
 language
of
functions.
 
 Definition:
Function
 A
function
is
a
rule
that
assigns
to
each
element
from
one
 set
exactly
one
element
from
another
set.
 
 
 For
example,
the
rule
that
pairs
each
student
in
the
class
with
 their
birthday
is
a
function,
since
every
student
has
one
and
 only
one
birthday.

However,
the
rule
that
pairs
each
day
of
the
 year
with
the
student
that
was
born
on
that
day
may
not
be.

 Why?
 The
notation
that
we
typically
use
when
working
with
a
 function
looks
like
 





 f ( x ) ,

read
“f
of
x”,
 
 where
 f 
is
the
name
of
the
function,
 x 
is
an
element
from
one
 set
(input),
and
the
value
of
 f ( x ) 
is
the
corresponding
element
 from
the
other
set
(output).
 
 For
example,
the
equation
 f ( 3) = 7 
indicates
that
for
the
 function
rule
named
 f ,
the
number
3
is
assigned
to
the
number
 7.


 Keep
in
mind,
it
is
still
okay
for
other
“input”
values
to
also
be
 paired
with
7,
so
 f ( −5 ) = 7 
can
also
be
true.
 In
most
cases,
the
rule
that
is
to
be
applied
is
written
as
a
 mathematical
expression.

For
example,
 
 




 f ( x) = x2 + 2x − 8
 
 indicates
that
for
the
function
named
 f 
the
rule
will
be
to
take
 the
input
number
 x 
square
it,
add
to
that
2
times
it,
and
then
 subtract
from
that
total
8
to
get
the
corresponding
output.
 
 So,
when
 x = 3,
we
get
 32 + 2 ( 3) − 8 = 9 + 6 − 8 = 7 .

This
implies
 that
 f ( 3) = 7 .
 
 2 Similarly,
 f ( −5 ) = ( −5 ) + 2 ( −5 ) − 8 = 25 − 10 − 8 = 7 .
 Example:
 For
each
function,
find

 (a)
 f ( 4 ) ,

 
 (b)
 f ( −1) ,

 
 1.

 f ( x ) = 2 x 2 − 5 x + 1
 
 
 
 x +1 2.

 f ( x ) = 
 x−2 
 
 
 3.

 f ( x ) = 3x −1 
 
 (c)
 f ( 2 a ) .
 Definition:

Domain
 The
set
of
all
“input”
values
of
a
function
is
called
its
 domain.


 
 For
a
general
function,
unless
otherwise
stated,
we
will
 assume
that
the
domain
is
the
largest
set
of
real
numbers
 for
which
the
expression
defining
the
function
is
defined.
 
 When
finding
the
domain
of
a
function,
we
must
avoid
 1)
division
by
zero
 2)
square
roots
(or
any
even
index)
of
a
negative
number
 3)
logarithms
of
a
non‐positive
number.
 
 Example:
 Determine
the
domain
of
the
function.
 
 
 
 
 1.

 f ( x ) = 3x + 7 
 
 
 
 x+3 
 
 
 2.

 f ( x ) = 2 
 x −4 
 
 
 x−2 
 
 
 3.

 f ( x ) = 2 
 x − 4x − 5 Definition:

Range
 The
set
of
all
“output”
values
of
a
function
is
called
its
 range.
 
 
 Generally,
it
is
difficult
to
determine
the
range
of
a
function
just
 from
the
expression
defining
it.

However,
if
we
have
a
graph
of
 that
function,
it
is
quite
easy
to
determine
by
examining
which
 y ‐values
are
attained.
 
 To
consider
the
graph
of
a
function,
we
need
to
identify
the
 value
of
the
function
 f ( x ) 
with
the
 y ‐coordinate
on
the
graph.

 That
is,
 y = f ( x ) .
 
 For
example,
if
we
consider
the
function
 f ( x ) = x 2 + 2 x − 8 ,
and
 make
a
table
of
values,
and
plot
those
points,
what
shape
 results?
 
 y = f ( x )
 x
 0
 ‐8
 1
 ‐5
 ‐1
 ‐9
 2
 0
 ‐2
 ‐8
 What
is
the
domain
and
range
of
the
function
shown
here?
 
 
 As
a
final
note,
as
the
definition
of
a
function
implies,
when
 graphing
 y = f ( x ) ,
for
each
 x 
in
the
domain
of
 f ,
there
will
be
 exactly
ONE
corresponding
value
of
 y .
 
 So,
if
we
draw
a
vertical
line
through
that
value
of
 x ,
it
will
 intersect
the
graph
of
 f 
exactly
one
time.

If
when
we
draw
a
 vertical
line
through
ANY
value
of
 x 
and
it
intersects
the
graph
 more
than
once,
this
cannot
be
the
graph
of
a
function
of
 x .
 
 What
does
it
mean
if
a
vertical
line
drawn
through
a
value
of
 x 
 doesn’t
intersect
the
graph
at
all?
 Clicker
Check­Out:

Press
“E”
to
check
out
now.
 
 
 Tomorrow
in
recitation:

Prerequisite
Skills
Exam
 
 
 NEXT
TIME
–
Section
3.1
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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