Section 1.5 Exponential functions
An exponential function is of the form f ( x )  a x for a positive number a.
If a>1, f ( x )  a x approaches 0 as x approaches negative infinity and grows rapidly upward as x
approaches positive infinity.
Graph 2 x in your calculator. Graph (1 2 ) x and see that the graph is a reflection across the y-axis of 2 x .
This is because (1 2 ) x = 2  x .
Exponent rules you should know are:
a
x y
x
 a a
y
a
x
1

x
a
 1 
  
 a 
x
a 
x
y
 a
x y
When the exponent is a rational number such as 1/4, we write a 1 4 
a
3 4

4
a
3


4
Example: 8 2 3 

3
a
3
.
8
2

3
64  4
or
8
2 3
x
4(2 )
Simplify the expression
x
4(2 )
1 
 
 4
x
2

2 2
x
 1 
 2 
 2 
x

2
1 
 
 4
x2
2 
2
x

x
.
2
2
x2
2x
 2
3x2

 8
3
2
 2
2
 4
4
a
, the 4th root of a. Also
Some applications of exponential functions are exponential growth and decay.
Example: The time required to eliminate half of the amount of a certain drug in the body is 4 days.
If 15 mg are present, how much is left in the body t days later?
t
Amount left in mg
4
15(1/2)
8
15(1/2)(1/2)
12
15(1/2)(1/2)(1/2) and so on. The factor 1/2 is occurs t/4 times.
A ( t )  15 (1 2 )
t 4
t
In general,
A ( t )  A0 ( 1 )
2
half
life
for exponential decay.
Next is an example of exponential growth.
Example: A culture weighs 2mg and the weight triples every 45 minutes. Find the amount present after t
hours. What is the weight after 3 hours?
Solution: The tripling time is 45 minutes = 45/60 hour = 3/4 hour.
A (t )  2 (3)
t
3/4
 2 (3)
4t 3
A(3)=2( 3 4 )=162 mg
Compound interest: If P dollars is invested at annual interest rate r compounded annually,
then the accumulated amount at the end of t years is
A ( t )  P (1  r )
t
is an exponential function of t.
If the interest is compounded m times per year then
r 

A(t)  P 1 

m 

mt
We can imagine that money can flow like a liquid. If interest is compounded infinitely often so the
interest continuously flows into the account, what is the accumulated amount? For this we need the
number e.
The number e: Graph
1 

Y 1  1 

x 

x
in windows where xmax is very large. You will see that the graph
approaches a horizontal line. The y-value on this line is the number e.
e = 2.71828… with never ending, non-repeating decimals.
Algebra then can show that
r 

1  
x


1
  1 


x r

x




x r




r
must approach
e
r
as x approaches infinity.
Back to continuous compound interest, if P dollars is invested at annual interest rate, r,
compounded continuously then
A ( t )  Pe
rt
is the accumulated amount after t years.
When the rate of growth of a substance or population is proportional to the amount present, the
amount is given as an exponential function similar to this one. Exponential functions grow rapidly
eventually so that even if r is small, the value will eventually exceed any given polynomial in t.
In fact for any positive r and any n,
t
e
n
rt
approaches 0 as t approaches positive infinity.