exponent. It should not be confused with the power function t▯x▯ ▯ x , in which the vari-
able is the base.
N In Appendix G we present an alternativeIn general, an exponential function is a function of the form
approach to the exponential and logarithmic
functions using integral calculus. f▯x▯ ▯ ax
where a is a positive constant. Let’s recall what this means.
MATH140 - Lecture 4 Notesve integer, then
a ▯ a ▯ a ▯ ▯▯▯ ▯ a
Exponential function is dependent on the increasing or decreasing function. In general,a positive integer, then
it is of the form f(x) = a^x where a is a positive constant. 1
a ▯n▯ n
To understand exponential functions, ﬁrst look at the graph of an exponential function.
If x is a rational number, x ▯ p▯q, where p and q are integers and q ▯ 0, then
y x p▯q q p q p
a ▯ a ▯ s a ▯ (s a)
But what is the meaning of a if x is an irrational number? For instance, what is meant by
2s3 or 5 ?
To help us answer this question we ﬁrst look at the graph of the function y ▯ 2 , where
x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the
1 domain of y ▯ 2 to include both rational and irrational numbers.
There are holes in the graph in Figure 1 corresponding to irrational values of x.We want
0 x x
1 to ﬁll in the holes by deﬁning f▯x▯ ▯ 2 , where x ▯ ▯, so that f is an increasing function.
In particular, since the irrational numse3 satisﬁes
Based on an understanding of the graph, we can use the concept of 1:1 to understand ▯ 1.8
whether a function is valid. What we mean by this 1:1 is that no two points go to the
same x value.
Deﬁnition: A one to one function is a function where every x has at most one y. No two
values of x give the same f(x) and no two values of y come from the same x. Formally,
f(x) is one to one if: x1 ≠ x2 㲗 f^-1(y) = x
Now that we know exponential functions are valid because they pass the 1:1 test, we
can examine their properties. Below is a graph of members of the family of functions y =
a^x where there are varying values of the base a. All the graphs pass through the same
point (0, 1) because a^0 = 1 for a ≠ 0. There are three essential points to exponential
• If 0 < a < 1, then the exponential function decreases
If a = 1, it is a constant
• If a > 1, it increases
Note, as the base a gets larger, the exponential function grows more rapidly (for x
> 0) Similary ▯ a about the y-axis. ia ▯ 0) where x is any irrational number. Figure 2
1 shows how all the holes in Figure 1 have been ﬁlled to complete the graph of the function
y f▯x▯ ▯ 2 , x ▯ ▯ y y
(00▯1) x The graphs of members of the family of funcy ▯ axare shown in Figure 3 for var-
ious valu0s of the base a. Notice that all of these graphs pass through the same point ▯0, 1▯
FIGURE 2 because a ▯ 1 fora ▯ 0 . Notice also that as the base a gets larger, the exponential func-
y=2®, x real (0,▯1) tion grows more rapidly (fx ▯10).
0 x 1 1 y 0 x
2▯▯▯’ ”4▯▯’ 10® 4® 2® 1.5®
0 x 0 x 0 x
(a) y=a®, ▯01
(a) y=a®, ▯01
FIGURE 4 NIf ▯ FIGURE 4enapprxacheasx
becomes large. , thenapproaches
as x decreases through negative values. In botOne reason for the importance of the exponential1®unction lies in the following proper-
cases the x-axis is a horizontal asymptote.ties. If x and y are rational numbers, then these laws are well known from elementaryper-
matters are discussed in Section 2.6.