Section 4.1 OnetoOne Functions
A function,
f
,is
onetoone
if
f
(
x
1
)=
f
(
x
2
) implies
x
1
=
x
2
, i.e., if
x
1
6
=
x
2
,then
f
(
x
1
)
6
=
f
(
x
2
).
Example.
The function
f
(
x
)=2
x
+1 is onetoone because
f
(
x
1
f
(
x
2
) implies 2
x
1
+1 = 2
x
2
+1
and this implies that
x
1
=
x
2
. The graph of
f
is shown below. Note that all horizontal lines intersects
the graph of
f
at one point.
–4
–2
2
4
–4
–2
2
4
f
(
x
x
Example.
f
(
x
x
2
is not onetoone. For example, if
x
1
=1and
x
2
=

1, then
x
1
6
=
x
2
but
f
(
x
1
f
(
x
2
). The graph of
f
is shown below. Note that some horizontal lines intersect the graph
of
f
at more than one point.
–4
–2
2
4
–4
–2
2
4
f
(
x
x
2
The above two graphs illustrate the following horizontal line test.
Horizontal Line Test.
A function
f
is onetoone iﬀ every horizontal line intersects the graph of
f
in at most one point.
A function
f
is
increasing
if
f
(
a
)
<f
(
b
) whenever
a<b
and
a
and
b
are in the domain of
f
.F
o
r
example, the function given by
f
(
x
)=3
x
+ 2 is an increasing function. A function
f
is
decreasing
if
f
(
a
)
>f
(
b
) whenever
and
a
and
b
are in the domain of
f
. For example, the function given
by
f
(
x

2
x
+ 3 is a decreasing function.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Kutter
 Algebra

Click to edit the document details