1.4 Exponential Functions
An exponential function is a function of the form f(x) = ax
where a is a positive constant. Consider the following cases:
1. If x = n, a positive integer, then:
a n = a  a  a  a
n factors
2. If x = 0, then a0 = 1.
3. If x = -n, where n is a positive integer, then:
a
−n
1
= n
a
4. If x is a rational number, x = p/q, where p and q are integers
and q > 0, then:
ax = a p/q = a p = ( a ) p
q
q
However, what is the meaning of ax if x is an irrational number?
For instance, what is meant by 2 3 or 2 ?
The graph of the function y = 2x, where x is rational, has
“holes”.
We want to enlarge the domain
of y = 2x to include both rational
and irrational numbers. We want
to fill in the holes. For example:
1.7  3  1.8 →
21.7  2
3
 21.8
1
Or we can continue with better approximations:
1.73  3  1.74
 21.73  2
3
1.732  3  1.733  21.732  2
 21.74
3
1.7320  3  1.7321  21.7320  2
1.73205  3  1.73206






 21.733
3
 21.7321
 21.73205  2 3  21.73206






It can be shown that there is exactly one number that is greater
than all the numbers
21.7, 21.73, 21.732, 21.7320, 21.73205, …
and less than all the numbers
21.8, 21.74, 21.733, 21.7321, 21.73206, …
We define 2
3
to be this number.
Similarly, we can define 2x (or ax, if a > 0) where x is any
irrational number. All the holes in the earlier figure have been
filled to complete the graph of the function f(x) = 2x, x  .
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The graphs of members of the family of functions y = ax are
shown here for various values of the base a.
Domain
and
range (0,∞) .
Ex.1. Sketch the graph of the function y = 4 − 3  2 x and
determine its domain and range.
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LAWS OF EXPONENTS
If a and b are positive numbers and x and y are any real
numbers, then:
1. ax + y = axay
2. ax – y = ax/ay
3. (ax)y = axy
4. (ab)x = axbx
The exponential function occurs very frequently in mathematical
models of nature and society. Of all possible bases for an
exponential function, there is one that is most convenient for the
purposes of calculus, number e. Some of the calculus formulas
will be greatly simplified if we choose the base a to be e. The
function f ( x) = e x is called the natural exponential function.
e ≈ 2.71828
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1
Ex.2. Find the domain of the functions: a) f ( x) = x2 − 4
;
e
−1
b) g ( x) = 8 − 2 x .
x
Ex.3. Find the exponential function f ( x) = Cb whose graph is
given below.
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Ex.4. Population growth. A bacteria culture starts with 100
bacteria and doubles in size every half hour. Find how many
bacteria are there after: (a) 3 hours? (b) t hours? (c) 40 minutes?
Radioactive decay. Radioactive isotopes such as Ca14 are used
to determine the age of fossils or minerals. This technique was
discovered about 100 years ago and is based on the property of
certain atoms to transform spontaneously by giving off protons,
neutrons or electrons. The phenomenon, called radioactive
decay occurs at a constant rate that is independent of
environmental conditions. The time required for the substance to
be reduced to half its initial values is called the half-life.
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Ex.5. The half-life of Sr90 is 25 years. If a sample of Sr90 has a
mass of 24mg, find an expression for the mass m(t) that remains
after t years.
It is common to write the radioactive decay law in the form
W (t ) = W0e− t , t  0
where   0 denotes the decay rate.
The more general formula
A(t ) = A0ekt , t  0
can be used to describe population growth (k>0) or radioactive
decay (k<0).
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