Graphics Reference
In-Depth Information
x
y
, thefunction is said to
be
one
-
one
and is called an
injection
. A function
which is both a surjection and an injection is
called a
bijection
.
Definition
A function is said to bea
real function
when
both its domain and its co-domain aresubsts
of R.
f
(
y
)
When
f
(
x
)
Theorem
qns 5, 6
If
f
:
A
B
is a bijection, then there is a
well-defined inverse function
f
:
B
A
such
that
f
(
f
(
a
))
a
, for all
a
A
.
Continuity
Throughout the eighteenth century, functions were given by
algebraic formulae, with variables being added, subtracted, multiplied or
divided, and combined with exponential, logarithmic or trigonometric
functions. Controversy surrounded the differential equation for a
plucked string, since in the ordinary starting position two straight-line
parts of the string meet at an angle, and thereby require different
algebraic formulae for different parts of the string. Cauchy took the
example of the function defined by
f
(
x
)
) to show that even
familiar algebraic expressions might lead to a graph with a sharp bend
in it.
(
x
7
Illustrateon a graph thefunctions
f
: R
R given by
(i)
f
(
x
)
0, when
x
0, and
f
(
x
)
x
when
x
0;
(ii)
f
(
x
)
0;
(iii)
f
(
x
)
x
when
x
0, and
f
(
x
)
x
when
x
0.
(iv)
f
(
x
)
0, when
x
0, and
f
(
x
)
x
when
x
x
ยท
x
.
Allowing a variety of formulae when defining a single function was
a significant first step towards a more general notion of function.
Further steps were taken when limiting processes were considered.
8
Examinethegraphs of
y
x
,
y
x
,
y
x
and
y
x
using a
computer or a graphic calculator.
For what values of
x
does lim
x
exist as
n
?
What is the largest subset
A
R for which a function
f
:
A
R may
be defined by
f
(
x
)
lim
x
as
n
?
What is therangeof this function? Sktch its graph.
x
1
9
Examinethegraphs of
y
1
, for
n
2, 3, 10, 11,
x
using a computer or a graphic calculator.