Even and odd functions
A function f
is called
even
provided that
=
(
)
f

x
(
)
f
x
for all numbers
x
in the domain of f.
If
a
is a positive number, an
even function
has the same value at a negative number

a
as it has at the corresponding positive number
a
.
An example of an even function is the function f given by
=
(
)
f
x
x
2
.
The graph of an even function is
symmetrical about the y axis
.
Given a point P (
)
,
a
(
)
f
a
on the graph, where
a
is positive, the point Q (
)
,

a
(
)
f
a
is also on the graph, and the
y
axis is a rightangle
bisector of the line QP, that is, QM = MP and angle QMO is 90 degrees.
Examples of even functions
.
Example 1
:
=
(
)
f
x
x
2
.
f is even because:
=
(
)
f

x
(
)

x
2
=
(
)

1
x
2
=
(
)

1
2
x
2
=
x
2
=
(
)
f
x
.
Example 2
:
=
(
)
f
x

x
4
4
x
2
.
f is even because:
=
(
)
f

x

(
)

x
4
4 (
)

x
2
=

x
4
4
x
2
=
(
)
f
x
.
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Example 3
:
=
(
)
f
x

4
x
2
.
f is even because:
=
(
)
f

x

4
(
)

x
2
=

4
x
2
=
(
)
f
x
.
Example 4
:
=
(
)
f
x

4
x
.
f is even because:
=
(
)
f

x

4

x
=

4
x
=
(
)
f
x
.
A function f is called
odd
provided that
=
(
)
f

x

(
)
f
x
for all numbers
x
in the domain of f.
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 Spring '08
 Uri
 Even and odd functions, QMO

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