Graphics Reference
In-Depth Information
(iv)
f
(
x
)
log
x
,
(vi)
f
(
x
)
(
x
4)/(
x
2) (beware of
x
2).
1/
x
, (v)
f
(
x
)
It is sometimes convenient to denote the range of the function
f
:
A
R by
f
(
A
) and to say that
f
maps
A
to
f
(
A
).
Although, for every function,
f
,
x
y
f
(
x
)
f
(
y
), it is only
sometimes the case that
f
(
x
)
f
(
y
)
x
y
. When this second
implication holds, thefunction
f
is said to be
one
-
to
-
one
or
one—one
, and
is called an
injection
.
4
Which of the functions given in questions 2 and 3 are one
—
one?
Bijections and inverse functions
If a function
f
:
A
B
, with domain
A
and co-domain
B
, is both
one
—
one
and
onto, not only is each element
a
A
matched with a
unique element
f
(
a
)
B
, which is, of course, true for any function
f
, but
also, for each element
b
B
, there is a unique element
a
A
such that
f
(
a
)
b
. A function which is both one
—
oneand onto is called a
bijection
. Such a function has an inverse
g
:
B
A
, defined by
g
(
f
(
a
))
a
. When this is the case, we will write
g
f
.
5
For each of the functions described in qns 2 and 3, identify
appropriatesubsts
A
,
B
R, such that
f
:
A
B
is a bijection.
6
Thepoints on thegraph of a real function
f
havetheform (
x
,
f
(
x
)).
If
f
:
A
B
is a bijction, why do thepoints on thegraph of
f
havetheform (
f
(
x
),
x
)? Sketch the graph of the function
f
given by
f
(
x
)
x
for positive
x
and sketch the graph of its inverse function
f
(
x
)
x
. Also sketch the graph of exp (the function
E
in
chapter 11) and its inverse, log.
Summary
-
functions
Definition
qns 1, 2,
3, 4,5
If a set
A
and a set
B
are given, then a
function
f
:
A
B
is a pairing of each element of
A
with
an element of
B
. Thest
A
is called the
domain
of thefunction. Thest
B
is called the
co
-
domain
of the function. The element of
B
which is
paired with
a
A
is denoted by
f
(
a
). Thest
f
(
a
)
a
A
B
is called the
range
of the
function. When the range of a function is the
wholeof its co-domain, thefunction is said to
be
onto
its co-domain and is called a
surjection
.
