2.3.a properties of functions
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Transcript of 2.3.a properties of functions
One-to-One Functions
A function is if and only
if whenever and are two numbers in the
domain of and th
one-to-one 1
e .
1
n
f
a b
f a b f a f b
2
One-to-One Functions
, then
Equivalently, is one-to-one if and only
if whenever .
One-to-one functions are also called
functioninjective s.
f a f b a b
f
3
Example 2.3.1
Determine if the following functions are
one-to-one or not by using the definition.
1. 3 7
3 7 3 7
3 3
Therefore, the function is 1-1.
y f x x
f a f b
a b
a b
a b
4
2 12.
2 1 2 1
2 1 2 1
2 2
Therefore, the function is 1-1.
xy g x
x
g a g b
a b
a b
a b b a
ab b ab a
b a
b a
5
Horizontal Line Test
every horizontal li
A function is one-to-one if and only if
intersects the graph
of the func
ne
at most ontio e pn o in .int
f
7
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
Example 2.3.2
Determine if the following functions are
one-to-one or not by using horizontal
line test.
1. 3 7y f x x
1-1
8
Onto Functions
A function from to , : is
called if for all there is an
onto
such that .
f A B f A B
b B
a A f a b
11
Onto Functions
Each element of has a in
under .
An onto function is also called .
pre-image
surjective
B
A f
12
Example 2.3.3
Determine if the following functions are onto
given the indicated sets.
1. :
where 3 7
. Therefore is onto.
f R R
f x x
Rng f R f
14
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
2. : 0 2
2 1where
2
Therefore is onto.
g R R
xg x
x
Rng g R
g
15
Example 2.3.4
Determine if the following functions are
bijective given the indicated sets.
1. :
where 3 7
Since is 1-1 and onto, is bijective.
f R R
f x x
f f
19
Increasing Functions
1 2 1 2
1 2
A function is on an interval
if and only if
whenever
where and are any numbers in the
interva
increas
l.
ingf
f x f x x x
x x
22
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
2Consider 4 on ,0f x x
1 2
1 2
If then x x
f x f x
24 is increasing on ,0f x x 23
Decreasing Functions
1 2 1 2
1 2
A function is on an interval
if and only if
whenever
where and are any numbers in the
interva
decreas
l.
ingf
f x f x x x
x x
24
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
2Consider 4 on 0,f x x
1 2
1 2
If then x x
f x f x
24 is decreasing on 0,f x x 25
Monotonic Functions
If a function is either increasing or decreasing
on an interval, then it is said to be
on the i
monotoni
nter
c
val.
26
Example 2.3.5
1 2
1 2
1 2
1 2
Show by definition that 3 7
is monotonic increasing in R.
3 3
3 7 3 7
Therefore, is increasing on R.
f x x
x x
x x
x x
f x f x
f27
Example 2.3.6
1 2
1 2
1 2
1 2
Show by definition that 5
is monotonic decreasing in R.
5 5
Therefore, is decreasing on R.
g x x
x x
x x
x x
g x g x
g28
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
Example 2.3.7 Consider the graph of . Determine if the function is monotonic increasingor decreasing on the indicated intervals.
f
a. 0,
b. , 1
c. 2,0
d. R29
Bounded Functions
If the range of a function is bounded
above, then the function is sai
bounded
d to
ab
be
ove.
30
Example 2.3.7
2
2
Determine if the following functions
are bounded above or not.
1. 1
0,1
is bounded above by 1.
2. 1
1,
is not bounded above.
f x x
Rng f
f
g x x
Rng g
g
31
Bounded Functions
If the range of a function is bounded
below, then the function is sai
bounded
d to
be
be
low.
32
Example 2.3.8
2
2
Determine if the following functions
are bounded below or not.
1. 1
0,1
is bounded below by 0.
2. 1
1,
is bounded below by 1.
f x x
Rng f
f
g x x
Rng g
g
33
Bounded Functions
If the range of the function is both
bounded above and below, the
function is said to boube nded.
34
Example 2.3.9
2
Determine if the following functions
are bounded or not.
1. 1
0,1
is bounded above by 1.
is bounded below by 0.
Therefore, is bounded.
f x x
Rng f
f
f
f
35