# MATH 140 Lecture Notes - Lecture 4: Exponential Function, Irrational Number, Rational Number

by OC13018

School

McGill UniversityDepartment

Mathematics & Statistics (Sci)Course Code

MATH 140Professor

Ewa DumaLecture

4 MATH140 - Lecture 4 Notes

Exponential Functions:

Exponential function is dependent on the increasing or decreasing function. In general,

it is of the form f(x) = a^x where a is a positive constant.

To understand exponential functions, ﬁrst look at the graph of an exponential function.

36. The ﬁrst graph in the ﬁgure is that of as displayed

by a TI-83 graphing calculator. It is inaccurate and so, to help

explain its appearance, we replot the curve in dot mode in the

second graph.

What two sine curves does the calculator appear to be plotting?

Show that each point on the graph of that the TI-

83 chooses to plot is in fact on one of these two curves. (The

TI-83’s graphing window is 95 pixels wide.)

y!sin 45x

0 2π0 2π

y!sin 45x

(b) Graph the function using the viewing rectangle

by . How does this graph differ from the

graph of the sine function?

35. The ﬁgure shows the graphs of and as

displayed by a TI-83 graphing calculator.

The ﬁrst graph is inaccurate. Explain why the two graphs

appear identical. [Hint: The TI-83’s graphing window is 95

pixels wide. What speciﬁc points does the calculator plot?]

y=sin96x

0 2π

y=sin2x

0 2π

y!sin 2xy !sin 96x

!!1.5, 1.5"!!5, 5"

y!sin#x2$

EXPONENTIAL FUNCTIONS

The function is called an exponential function because the variable, x, is the

exponent. It should not be confused with the power function , in which the vari-

able is the base.

In general, an exponential function is a function of the form

where is a positive constant. Let’s recall what this means.

If , a positive integer, then

nfactors

If , and if , where is a positive integer, then

If is a rational number, , where and are integers and , then

But what is the meaning of if xis an irrational number? For instance, what is meant by

or ?

To help us answer this question we ﬁrst look at the graph of the function , where

xis rational. A representation of this graph is shown in Figure 1. We want to enlarge the

domain of to include both rational and irrational numbers.

There are holes in the graph in Figure 1 corresponding to irrational values of x. We want

to ﬁll in the holes by deﬁning , where , so that is an increasing function.

In particular, since the irrational number satisﬁes

1.7 "s3 "1.8

s3

fx !!f#x$!2x

y!2x

y!2x

5

#

2s3

ax

ax!ap%q!q

sap!

(

q

sa

)

p

q$0qpx !p%qx

a!n!1

an

nx !!nx !0, then a0!1

an!a"a"%%% "a

x!n

a

f#x$!ax

t#x$!x2

f#x$!2x

1.5

52

||| |

CHAPTER 1 FUNCTIONS AND MODELS

FIGURE 1

Representation of y=2®, x rationa

l

x

0

y

1

1

NIn Appendix G we present an alternative

approach to the exponential and logarithmic

functions using integral calculus.

Based on an understanding of the graph, we can use the concept of 1:1 to understand

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 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

functions:

•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)

SECTION 1.5 EXPONENTIAL FUNCTIONS

||||

53

we must have

and we know what and mean because 1.7 and 1.8 are rational numbers. Similarly,

if we use better approximations for , we obtain better approximations for :

. . . .

. . . .

. . . .

It can be shown that there is exactly one number that is greater than all of the numbers

...

and less than all of the numbers

...

We deﬁne to be this number. Using the preceding approximation process we can com-

pute it correct to six decimal places:

Similarly, we can deﬁne (or , if ) where xis any irrational number. Figure 2

shows how all the holes in Figure 1 have been ﬁlled to complete the graph of the function

.

The graphs of members of the family of functions are shown in Figure 3 for var-

ious values of the base a. Notice that all of these graphs pass through the same point

because for . Notice also that as the base agets larger, the exponential func-

tion grows more rapidly (for ).

You can see from Figure 3 that there are basically three kinds of exponential functions

. If , the exponential function decreases; if , it is a constant; and if

, it increases. These three cases are illustrated in Figure 4. Observe that if ,a!1a!1

a"10 "a"1y"ax

FIGURE 3 0

1®

1.5®

2®

4®

10®

”’®

1

4

”’®

1

2

x

y

1

x!0

a!0a0"1

!0, 1"

y"ax

f!x""2x, x!!

a!0ax

2x

2s3#3.321997

2s3

21.73206, 21.7321, 21.733, 21.74, 21.8,

21.73205, 21.7320, 21.732, 21.73, 21.7,

1.73205 "s3 "1.73206 ?21.73205 "2s3"21.73206

1.7320 "s3 "1.7321 ? 21.7320 "2s3"21.7321

1.732 "s3 "1.733 ? 21.732 "2s3"21.733

1.73 "s3 "1.74 ? 21.73 "2s3"21.74

2s3

s3

21.8

21.7

21.7 "2s3"21.8

x

1

0

y

1

F IGURE 2

y

=2®, x real

NA proof of this fact is given in J. Marsden

and A. Weinstein,

Calculus Unlimited

(Menlo

Park, CA: Benjamin/Cummings, 1981). For an

online version, see

www.cds.caltech.edu/~marsden/

volume/cu/CU.pdf

NIf , then approaches as

becomes large. If , then approaches

as decreases through negative values. In both

cases the -axis is a horizontal asymptote. These

matters are discussed in Section 2.6.

x

x

0ax

a!1

x0ax

0"a"1

Laws of Exponents:

If a and b are positive numbers and x and y are any real numbers, then there are four

rules to consider:

then the exponential function has domain !and range . Notice also that,

since , the graph of is just the reﬂection of the graph of

about the -axis.

One reason for the importance of the exponential function lies in the following proper-

ties. If xand yare rational numbers, then these laws are well known from elementary

algebra. It can be proved that they remain true for arbitrary real numbers xand y. (See

Appendix G.)

LAWS OF EXPONENTS If aand bare positive numbers and xand yare any real

numbers, then

1. 2.3. 4.

EXAMPLE 1 Sketch the graph of the function and determine its domain and

range.

SOLUTION First we reﬂect the graph of [shown in Figures 2 and 5(a)] about the

x-axis to get the graph of in Figure 5(b). Then we shift the graph of

upward 3 units to obtain the graph of in Figure 5(c). The domain is !and

the range is .

M

EXAMPLE 2 Use a graphing device to compare the exponential function

and the power function . Which function grows more quickly when xis large?

SOLUTION Figure 6 shows both functions graphed in the viewing rectangle

by . We see that the graphs intersect three times, but for the graph of x!4!0, 40"

!"2, 6"

t#x$!x2

f#x$!2x

V

F I G U RE 5

0

1

(a) y=2®

x

y

0

_1

(b) y=_2®

x

y

y=3

0

2

(c) y=3-2®

x

y

#"#, 3$

y!3"2x

y!"2x

y!"2x

y!2x

y!3"2x

#ab$x!axbx

#ax$y!axy

ax"y!ax

ay

ax$y!axay

1(0,1)

(a) y=a®, 0<a<1 (b) y=1® (c) y=a®, a>1

(0,1)

FIGURE 4

x

0

y

x

0

y

x

0

y

yy !ax

y!#1%a$x

#1%a$x!1%ax!a"x

#0, #$y!ax

54

||||

CHAPTER 1 FUNCTIONS AND MODELS

www.stewartcalculus.com

For review and practice using the Laws of

Exponents, click on Review of Algebra.

NFor a review of reﬂecting and shifting graphs,

see Section 1.3.

Examples of exponential functions:

then the exponential function has domain !and range . Notice also that,

since , the graph of is just the reﬂection of the graph of

about the -axis.

One reason for the importance of the exponential function lies in the following proper-

ties. If xand yare rational numbers, then these laws are well known from elementary

algebra. It can be proved that they remain true for arbitrary real numbers xand y. (See

Appendix G.)

LAWS OF EXPONENTS If aand bare positive numbers and xand yare any real

numbers, then

1. 2.3. 4.

EXAMPLE 1 Sketch the graph of the function and determine its domain and

range.

SOLUTION First we reﬂect the graph of [shown in Figures 2 and 5(a)] about the

x-axis to get the graph of in Figure 5(b). Then we shift the graph of

upward 3 units to obtain the graph of in Figure 5(c). The domain is !and

the range is .

M

EXAMPLE 2 Use a graphing device to compare the exponential function

and the power function . Which function grows more quickly when xis large?

SOLUTION Figure 6 shows both functions graphed in the viewing rectangle

by . We see that the graphs intersect three times, but for the graph of x!4!0, 40"

!"2, 6"

t#x$!x2

f#x$!2x

V

F I G U R E 5

0

1

(a) y=2®

x

y

0

_1

(b) y=_2®

x

y

y=3

0

2

(c) y=3-2®

x

y

#"#, 3$

y!3"2x

y!"2x

y!"2x

y!2x

y!3"2x

#ab$x!axbx

#ax$y!axy

ax"y!ax

ay

ax$y!axay

1(0,1)

(a) y=a®, 0<a<1 (b) y=1® (c) y=a®, a>1

(0,1)

FIGURE 4

x

0

y

x

0

y

x

0

y

yy !ax

y!#1%a$x

#1%a$x!1%ax!a"x

#0, #$y!ax

54

||| |

CHAPTER 1 FUNCTIONS AND MODELS

www.stewartcalculus.com

For review and practice using the Laws of

Exponents, click on Review of Algebra.

NFor a review of reﬂecting and shifting graphs,

see Section 1.3.

Domain of a function:

Given a function f(x), its domain is the set of values of x for which it is deﬁned. Below

are some common domains:

•The domain of any polynomial is ANY REAL NUMBERS (-inf, inf)

•The domain of sqrt(f(x)) is the set of x values such that f(x) ≥ 0

•The domain of 1/f(x) is the set of x values such that f(x) ≠ 0

•The domain of arccos(x) and arcsin(x) is [-1, 1]

•The domain of sin(x) and cos(x) is ANY REAL NUMBERS

To ﬁnd the domain of a given function, you might have to combine different domains to

get the ﬁnal result.

Example: Find the domain of f(x) = sqrt(x-2)/(x^2-9)

For the numerator we want x ≥ 2

For the fraction we want x^2 - 9 ≠ 0 > x ≠ 3, x ≠ -3 therefore… the domain is [2,3) U (3,

inf)

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