Download What is the domain of an exponential function?

Let’s examine exponential functions. They are
different than any of the other types of functions we’ve
studied because the independent variable is in the
exponent.
x
3
2
1
0
-1
-2
-3
2x
8
4
2
1
1/2
1/4
1/8
f x   2
Let’s look at the graph of
this function by plotting
x some points.
8
BASE
Recall what a
negative exponent
means:
f  1  2 1 
1
2
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
Compare the graphs 2x, 3x , and 4x
Characteristics about the
Graph of an Exponential
x
Function f x   a where a > 1
1. Domain is all real numbers
2. Range is positive real numbers
3. There are no x intercepts because
there is no x value that you can put
in the function to make it = 0
4. The y intercept is always (0,1)
because a 0 = 1
5. The graph is always increasing
6. The x-axis (where y = 0) is a
horizontal asymptote for x  - 
f x   4 x
f x   3x
f x   2 x
Can
What
What
you
is
isthe
the
seerange
x
What
Are
these
is
the
ythe
of
intercept
horizontal
domain
an exponential
of
of
these
intercept
exponential
ofan
these
function?
exponential
asymptote
exponential
for
exponential
functions
functions?
these
function?
functions?
functions?
increasing
or
decreasing?
y2
x
All of the transformations that you
learned apply to all functions, so what
x
would the graph of y  2  3
look like?
up 3
right 2 down 1
up 1
Reflected over
x axis
y  1 2
x
y2
x2
1
Reflected about y-axis
y2
x
This equation could be rewritten in
a different form:
x
y2
x
1 1
 x  
2
2
So if the base of our exponential
function is between 0 and 1
(which will be a fraction), the
graph will be decreasing. It will
have the same domain, range,
intercepts, and asymptote.
There are many occurrences in nature that can be
modeled with an exponential function. To model these
we need to learn about a special base.
The Natural Base : e
Instead of using base as a number, in
application problems, we can use base e.
• e : Natural base
• e = 2.718281828…..
• You need to remember this value
The Base “e” (also called the natural base)
To model things in nature, we’ll
need a base that turns out to be
between 2 and 3. Your calculator
knows this base. Ask your
calculator to find e1. You do this by
using the ex button (generally you’ll
need to hit the 2nd or yellow button
first to get it depending on the
calculator). After hitting the ex, you
then enter the exponent you want
(in this case 1) and push = or enter.
If you have a scientific calculator
that doesn’t graph you may have to
enter the 1 before hitting the ex. Example
You should get 2.718281828
for TI-83
•
•
•
•
•
•
•
•
Well, let me show you how to remember:
Remember to start with 2.
Who was the 7th president of US?
ANDREW JACKSON
When was he elected?
Make a square with sides 1828.
Write 1828 twice.
Make a diagonal. What type of triangle is
this?
• And so on………I am tired.
• 2.718281828459045…….
f x   e x
f x   3x
f x   2 x
The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many
applied exponential functions. This irrational number is approximately equal to
2.72. More accurately,
The number e is called the natural
e base. The function f (x) = ex is called the
2.71828...
natural exponential function.
f (x) = 3x f (x) = ex
4
f (x) = 2x
(1, 3)
3
(1, e)
2
(1, 2)
(0, 1)
-1
1
Translations:
Application:
Formulas for Compound Interest
After t years, the balance, A, in an account with
principal P and annual interest rate r (in
decimal form) is given by the following
formulas:
nt
 r
1. For n compounding per year: A  P  1  

2. For continuous compounding: A = Pert.
n
Example: Choosing Between Investments
You want to invest $8000 for 6 years, and you have a choice between two
accounts. The first pays 7% per year, compounded monthly. The second
pays 6.85% per year, compounded continuously. Which is the better
investment?
Solution The better investment is the one with the greater balance in
the account after 6 years. Let’s begin with the account with monthly
compounding. We use the compound interest model with P = 8000,
r = 7% = 0.07, n = 12 (monthly compounding, means 12 compounding's
per year), and t = 6.
 
r
A  P 1
n
nt

0.07
 8000 1 
12

12*6
 12,160.84
The balance in this account after 6 years is $12,160.84.
You want to invest $8000 for 6 years, and you have a choice between two
accounts. The first pays 7% per year, compounded monthly. The second pays
6.85% per year, compounded continuously. Which is the better investment?
Solution For the second investment option, we use the model for
continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6.
A  Pe rt  8000e0.0685(6)  12,066.60
The balance in this account after 6 years is $12,066.60, slightly less than
the previous amount. Thus, the better investment is the 7% monthly
compounding option.
Example
Use A= Pert to solve the following problem:
Find the accumulated value of an investment of
$2000 for 8 years at an interest rate of 7% if
the money is compounded continuously
Solution:
A= Pert
A = 2000e(.07)(8)
A = 2000 e(.56)
A = 2000 * 1.75
A = $3500
Try: