7.1 - CHAPTER 7 Some real numbers satisfy polynomial...

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 ONE-TO-ONE FUNCTIONS; INVERSES
One-to-One 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
absolute-value 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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CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS
Functions for which this kind of repetition does not occur are called one-to-one
functions .
DEFINITION 7.1.1
A function f is said to be one-to-one 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 one-to-one 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 one-to-one. The cubing function is one-to-one because no two distinct numbers
have the same cube. The square-root function is one-to-one because no two distinct
nonnegative numbers have the same square root.
x
y
x
2
f ( x
1
) = f ( x
2
) f is not one-to-one:
x
1
Figure 7.1.1
x
y
f is one-to-one:
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 one-to-one. Look at the graph of the function. If some
horizontal line intersects the graph more than once, then the function is not one-to-one
(Figure 7.1.1). If, on the other hand, no horizontal line intersects the graph more than
once, then the function is one-to-one (Figure 7.1.2).
Inverses
We begin with a theorem about one-to-one functions.
THEOREM 7.1.2
If f is a one-to-one 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 .