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  • Let b = 1. Then f ( x ) = 1 x = 1 for any value of x .

To evaluate an exponential function with the form f ( x ) = b x , we simply substitute x with the given value, and calculate the resulting power. For example:

Let f ( x ) = 2 x . What is f ( 3 ) ?

f ( x ) = 2 x f ( 3 ) = 2 3   Substitute  x = 3. = 8   Evaluate the power .

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:

Let f ( x ) = 30 ( 2 ) x . What is f ( 3 ) ?

f ( x ) = 30 ( 2 ) x f ( 3 ) = 30 ( 2 ) 3 Substitute  x = 3. = 30 ( 8 )   Simplify the power first . = 240 Multiply .

Note that if the order of operations were not followed, the result would be incorrect:

f ( 3 ) = 30 ( 2 ) 3 60 3 = 216,000

Evaluating exponential functions

Let f ( x ) = 5 ( 3 ) x + 1 . Evaluate f ( 2 ) without using a calculator.

Follow the order of operations. Be sure to pay attention to the parentheses.

f ( x ) = 5 ( 3 ) x + 1 f ( 2 ) = 5 ( 3 ) 2 + 1 Substitute  x = 2. = 5 ( 3 ) 3 Add the exponents . = 5 ( 27 ) Simplify the power . = 135 Multiply .

Let f ( x ) = 8 ( 1.2 ) x 5 . Evaluate f ( 3 ) using a calculator. Round to four decimal places.

5.5556

Defining exponential growth

Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.

Exponential growth

A function that models exponential growth    grows by a rate proportional to the amount present. For any real number x and any positive real numbers a   and b such that b 1 , an exponential growth function has the form

  f ( x ) = a b x

where

  • a is the initial or starting value of the function.
  • b is the growth factor or growth multiplier per unit x .

In more general terms, we have an exponential function , in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let’s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function A ( x ) = 100 + 50 x . Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function B ( x ) = 100 ( 1 + 0.5 ) x .

A few years of growth for these companies are illustrated in [link] .

Year, x Stores, Company A Stores, Company B
0 100 + 50 ( 0 ) = 100 100 ( 1 + 0.5 ) 0 = 100
1 100 + 50 ( 1 ) = 150 100 ( 1 + 0.5 ) 1 = 150
2 100 + 50 ( 2 ) = 200 100 ( 1 + 0.5 ) 2 = 225
3 100 + 50 ( 3 ) = 250 100 ( 1 + 0.5 ) 3 = 337.5
x A ( x ) = 100 + 50 x B ( x ) = 100 ( 1 + 0.5 ) x

The graphs comparing the number of stores for each company over a five-year period are shown in [link] . We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.

Graph of Companies A and B’s functions, which values are found in the previous table.
The graph shows the numbers of stores Companies A and B opened over a five-year period.

Notice that the domain for both functions is [ 0 , ) , and the range for both functions is [ 100 , ) . After year 1, Company B always has more stores than Company A.

Now we will turn our attention to the function representing the number of stores for Company B, B ( x ) = 100 ( 1 + 0.5 ) x . In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and 1 + 0.5 = 1.5 represents the growth factor. Generalizing further, we can write this function as B ( x ) = 100 ( 1.5 ) x , where 100 is the initial value, 1.5 is called the base , and x is called the exponent .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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