1825_Sept29_notes_part1

1825_Sept29_notes_part1 - MTH
1825
sec.
006
 


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Unformatted text preview: MTH
1825
sec.
006
 
 Wednesday,
Sept.
29,
2010
 
 Section
2.6
–
Introduction
to
Functions
 
 
 Definition
of
a
function:
 
 Given
a
relation
in
x
and
y,
we
say
“y
is
a
function
of
x”
if
_______________________________________
 
 ________________________________________________________________________________________________________
 
 
 
 This
definition
means
that
if
two
ordered
pairs
have
the
same
first
coordinates
and
 different
second
coordinates,
then
________________________________________________________________
 
 
 Example:

Determine
whether
each
of
the
following
relations
is
a
function:
 
 a.
 
 b.
 {( 3, 7), (2, 7), (5, 7), ( 4, 7)} 
 
 {(6, 1), (5, 3), ( 7, 8), (5, 2)}
 
 
 
 €
 € 
 
 
 
 
 
 c.
 
 
 
 
 
 
 d.
 
 6
 5
 3
 
 
 
 
 
 2
 1
 4
 
 
 
 
 
 9
 2
 7
 
 
 
 
 
 
 3
 
 
 
 1
 
 
 Page
1
of
7
 MTH
1825
sec.
006
 
 Wednesday,
Sept.
29,
2010
 
 Vertical
Line
Test
 
 Consider
a
relation
defined
by
a
set
of
points
(x,
y)
in
a
rectangular
coordinate
system.

The
 graph
defines
y
as
a
function
of
x
if
_______________________________________________________________
 _______________________________________________________________________________________________________
 
 
 
 
 The
vertical
line
test
works
because
a
 
 vertical
line
represents
a
particular
x
value
 
 paired
with
every
possible
y
value.

If
it
 
 intersects
a
graph
more
than
once,
then
all
 
 the
intersection
points
have
the
same
x‐ 
 coordinate
but
different
y‐coordinates.
 
 
 
 
 x
=
3
 
 
 Example:

Determine
if
each
of
the
following
relations
define
y
as
a
function
of
x.
 
 
 
 
 
 
 
 
 
 
 
 Function
Notation
 
 Consider
the
equation
y
=
x
+
3.

This
is
the
set
of
all
ordered
pairs
such
that
the
y‐value
is
3
 more
than
the
x‐value.


 
 We
can
define
this
relationship
as
the
function
 f ( x ) = x + 3
where
f
is
the
name
of
the
 function,
x
is
the
input
value
from
the
domain
of
the
function,
and
 f ( x ) 
is
the
function
 value
(or
y‐value)
corresponding
to
x.

For
this
example,
 f (2) 
=
___________________.

The
point
 € ___________
is
on
the
graph
of
 y = f ( x ) .
 € € 
 € 
 Page
2
of
7
 MTH
1825
sec.
006
 
 Wednesday,
Sept.
29,
2010
 
 Example:

Use
the
functions
f,
g,
h,
and
k
below
to
find
the
listed
function
values.
 
 f ( x) = 8 x − 3 
 
 g( x ) = 12 
 h( x ) = − x 2 + 6 x − 3 
 
 
 k( x) = x − 5 
 
 
 f (−5) 
 
 a.
 
 
 
 
 b.
 h (−5) 
 € € € 
 
 
 € €
 
 
 
 
 g(−5) 
 
 c.
 
 
 
 
 d.
 k (−5) 
 
 
 
 € €
 
 
 g(π ) 
 
 e.
 
 
 
 
 f.
 k ( 7.3) 
 
 
 
 € €
 
 
 g.
 
 
 
 
 h.
 k ( m) 
 
 h (− w ) 
 
 
 
 €
 € 
 
 
 f ( x + 2) 
 f ( x − c) 
 i.
 
 
 
 
 j.
 
 
 
 € € € 
 
 Page
3
of
7
 MTH
1825
sec.
006
 
 Wednesday,
Sept.
29,
2010
 
 Finding
function
values
from
a
graph
 
 Example:

The
graph
of
 y = f ( x ) 
is
given.
 
 a.
 Find
 f (−5) .
 
 y = f ( x) 
 € 
 
 b.
 € Find
 f (1) .
 € 
 
 
 c.
 € For
what
values
of
x
is
 f ( x ) = −3
?
 
 
 
 € 
 d.
 For
what
values
of
x
is
 f ( x ) = 4 
?
 
 
 
 € 
 e.
 Write
the
domain
of
f

(in
interval
notation).
 
 
 f.
 Write
the
range
of
f

(in
interval
notation).
 
 
 
 Domain
of
a
Function
 
 The
domain
of
a
function
is
the
set
of
all
inputs
that
produce
a
real
value
for
the
output
 when
substituted
into
the
function.
 
 In
MTH
1825,
the
output
of
a
function
will
be
non‐real
when
one
of
two
things
happens:
 
 1.
 
 
 2.
 
 
 Any
value
of
x
that
causes
one
of
those
two
things
to
happen
will
be
excluded
from
the
 domain.

If
neither
of
those
two
things
happens,
the
domain
is
___________________.
 
 
 Page
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 ...
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This note was uploaded on 11/06/2010 for the course MTH 1835 taught by Professor Cioni during the Fall '10 term at Michigan State University.

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