CHAPTER
7
THE
TRANSCENDENTAL
FUNCTIONS
Some real numbers satisfy polynomial equations with integer coefficients:
3
5
satisfies the equation 5
x
−
3
=
0,
√
2 satisfies the equation
x
2
−
2
=
0.
Such numbers are called
algebraic.
There are, however, numbers that are not algebraic,
among them
π
. Such numbers are called
transcendental
.
Some functions
f
satisfy polynomial equations with polynomial coefficients:
f
(
x
)
=
x
π
x
+
√
2
satisfies the equation
(
π
x
+
√
2)
f
(
x
)
−
x
=
0,
f
(
x
)
=
2
√
x
−
3
x
2
satisfies the equation
[
f
(
x
)]
2
+
6
x
2
f
(
x
)
+
(9
x
4
−
4
x
)
=
0.
Such functions are called
algebraic
.There are, however, functions that are not algebraic;
these are called
transcendental
. For example, the trigonometric functions are tran
scendental functions. In this chapter we introduce other transcendental functions: the
logarithm function, the exponential function, and the inverse trigonometric functions.
But first, a little more on functions in general.
7.1
ONETOONE FUNCTIONS; INVERSES
OnetoOne Functions
A function can take on the same value at different points of its domain. Constant
functions, for example, take on the same value at all points of their domains. The
quadratic function
f
(
x
)
=
x
2
takes on the same value at
−
c
as it does at
c
; so does the
absolutevalue function
g
(
x
)
= 
x

. The function
f
(
x
)
=
1
+
(
x
−
3)(
x
−
5)
takes on the same value at
x
=
5 as it does at
x
=
3:
f
(3)
=
1,
f
(5)
=
1.
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372
CHAPTER 7
THE TRANSCENDENTAL FUNCTIONS
Functions for which this kind of repetition
does not
occur are called
onetoone
functions
.
DEFINITION 7.1.1
A function
f
is said to be
onetoone
if there are no two distinct numbers in
the domain of
f
at which
f
takes on the same value:
f
(
x
1
)
=
f
(
x
2
)
implies
x
1
=
x
2
.
Thus, if
f
is onetoone and
x
1
,
x
2
are different points of the domain, then
f
(
x
1
)
=
f
(
x
2
).
The functions
f
(
x
)
=
x
3
and
f
(
x
)
=
√
x
are both onetoone. The cubing function is onetoone because no two distinct numbers
have the same cube. The squareroot function is onetoone because no two distinct
nonnegative numbers have the same square root.
x
y
x
2
f
(
x
1
) =
f
(
x
2
)
f
is not onetoone:
x
1
Figure 7.1.1
x
y
f
is onetoone:
Figure 7.1.2
There is a simple geometric test, called the
horizontal line test,
which can be used to
determine whether a function is onetoone. Look at the graph of the function. If some
horizontal line intersects the graph more than once, then the function is not onetoone
(Figure 7.1.1). If, on the other hand, no horizontal line intersects the graph more than
once, then the function is onetoone (Figure 7.1.2).
Inverses
We begin with a theorem about onetoone functions.
THEOREM 7.1.2
If
f
is a onetoone function, then there is one and only one function
g
with
domain equal to the range of
f
that satisfies the equation
f
(
g
(
x
))
=
x
for all
x
in the range of
f
.
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 Real Numbers, Equations, Derivative, Inverse function, 1, Inverses

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