College Algebra 9th txtbk

332 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Table 1
a1 1 x
x
x b
1 2
10 2.593 74 …
100 2.704 81 …
1,000 2.716 92 …
10,000 2.718 14 …
100,000 2.718 27 …
1,000,000 2.718 28 …
By calculating the value of [1 (1x)] x for larger and larger values of x (Table 1), it
looks like [1 (1x)] x approaches a number close to 2.7183. In a calculus course, we can
show that as x increases without bound, the value of [1 (1x)] x approaches an irrational
number that we call e. Just as irrational numbers such as and 12 have unending, nonrepeating
decimal representations, e also has an unending, nonrepeating decimal representation.
To 12 decimal places,
e 2.718 281 828 459
2 1 0 1 2 3 4
Don’t let the symbol “e” intimidate you! It’s just a number.
Exactly who discovered e is still being debated. It is named after the great Swiss mathematician
Leonhard Euler (1707–1783), who computed e to 23 decimal places using
[1 (1x)] x .
The constant e turns out to be an ideal base for an exponential function because in calculus
and higher mathematics many operations take on their simplest form using this base.
This is why you will see e used extensively in expressions and formulas that model realworld
phenomena.
2
e
Z DEFINITION 2 Exponential Function with Base e
20
y
For x a real number, the equation
f (x) e x
10
defines the exponential function with base e.
y e x
y e x
x
5
5
Z Figure 5 Exponential functions.
The exponential function with base e is used so frequently that it is often referred to
as the exponential function. The graphs of y e x and y e x are shown in Figure 5.
EXAMPLE 3 Analyzing a Graph
Let g(x) 4 e x2 . Use transformations to explain how the graph of g is related to the
graph of f 1 (x) e x . Determine whether g is increasing or decreasing, find any asymptotes,
and sketch the graph of g.
SOLUTION
The graph of g can be obtained from the graph of f 1 by a sequence of three transformations:
f 1 (x) e x S f 2 (x) e x2 S f 3 (x) e x2 S g(x) 4 e x2
Horizontal Reflection Vertical
stretch in x axis translation
[See Fig. 6(a) for the graphs of f 1 , f 2 , and f 3 , and Fig. 6(b) for the graph of g.] The function
g is decreasing for all x. Because e x2 S 0 as x S , it follows that g(x) 4 e x2 S 4
as x S . Therefore, the line y 4 is a horizontal asymptote [indicated by the dashed
line in Fig. 6(b)]; there are no vertical asymptotes. [To check that the graph of g (as obtained
by graph transformations) is correct, plot a few points.]