# 6.1 Exponential functions  (Page 3/16)

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• Let $\text{\hspace{0.17em}}b=1.\text{\hspace{0.17em}}$ Then $\text{\hspace{0.17em}}f\left(x\right)={1}^{x}=1\text{\hspace{0.17em}}$ for any value of $\text{\hspace{0.17em}}x.$

To evaluate an exponential function with the form $\text{\hspace{0.17em}}f\left(x\right)={b}^{x},$ we simply substitute $x\text{\hspace{0.17em}}$ with the given value, and calculate the resulting power. For example:

Let $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}.\text{\hspace{0.17em}}$ What is $f\left(3\right)?$

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 $\text{\hspace{0.17em}}f\left(x\right)=30{\left(2\right)}^{x}.\text{\hspace{0.17em}}$ What is $\text{\hspace{0.17em}}f\left(3\right)?$

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

$f\left(3\right)=30{\left(2\right)}^{3}\ne {60}^{3}=216,000$

## Evaluating exponential functions

Let $\text{\hspace{0.17em}}f\left(x\right)=5{\left(3\right)}^{x+1}.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(2\right)\text{\hspace{0.17em}}$ without using a calculator.

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

Let $f\left(x\right)=8{\left(1.2\right)}^{x-5}.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(3\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and any positive real numbers and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}b\ne 1,$ an exponential growth function has the form

where

• $a\text{\hspace{0.17em}}$ is the initial or starting value of the function.
• $b\text{\hspace{0.17em}}$ is the growth factor or growth multiplier per unit $\text{\hspace{0.17em}}x$ .

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 $\text{\hspace{0.17em}}A\left(x\right)=100+50x.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}B\left(x\right)=100{\left(1+0.5\right)}^{x}.$

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

Year, $x$ Stores, Company A Stores, Company B
$0$ $100+50\left(0\right)=100$ $100{\left(1+0.5\right)}^{0}=100$
$1$ $100+50\left(1\right)=150$ $100{\left(1+0.5\right)}^{1}=150$
$2$ $100+50\left(2\right)=200$ $100{\left(1+0.5\right)}^{2}=225$
$3$ $100+50\left(3\right)=250$ $100{\left(1+0.5\right)}^{3}=337.5$
$x$ $A\left(x\right)=100+50x$ $B\left(x\right)=100{\left(1+0.5\right)}^{x}$

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.

Notice that the domain for both functions is $\text{\hspace{0.17em}}\left[0,\infty \right),$ and the range for both functions is $\text{\hspace{0.17em}}\left[100,\infty \right).\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}B\left(x\right)=100{\left(1+0.5\right)}^{x}.\text{\hspace{0.17em}}$ In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and $\text{\hspace{0.17em}}1+0.5=1.5\text{\hspace{0.17em}}$ represents the growth factor. Generalizing further, we can write this function as $\text{\hspace{0.17em}}B\left(x\right)=100{\left(1.5\right)}^{x},$ where 100 is the initial value, $\text{\hspace{0.17em}}1.5\text{\hspace{0.17em}}$ is called the base , and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is called the exponent .

what is Economics?
Is the study of human behaviour as a relationship between ends and scares mean which have alternative use
Alhaji
yes
Tawa
what is monopoly
Alhaji
what is labour
LABOUR is a measure of work done by human being
Blessing
It is all form of human effort use to utilize in production
Alhaji
Why is scarcity a foundermental problem in economics
Alhaji
Why is scarcity a foundermental problem in economics
scarcity occur unbalance demand and supply at this time cost goods increase then inflation very increase
Tesfaye
scarcity is a foundermental problem because its a natural situation and it affects the world at Large.in other words,it's limit in supply relating to deman
Akwosih
'Economics is about making choices in the presence of scarcity"
. 'Economics is about making choices in the presence of scarcity" - Dscuss.
manoj
describe the producer's scarce resources.. I.e land,Labour,capital and enterprise
short in supply
Charles
What are human behaviour?
the rationality in decision making
Charles
how can you describe economic goods in a much better easier way?
any thing that have utility
Charles
what is deman and supply
Demand can be defined as the ability and willingness to buy commodities in a given price of goods and services in a particular period of time
Alasana
supply refers to the ability and willingness to offered commodities for sale in a given price of goods and services in a period of time .
Alasana
Demand can refer to the ability and willingness to purchase a commodity at a giving price and time.
habib
what must the producer do if total costs exceed total revenue
raise price
Nguyen
reduce cost
Charles
scarcity resources sample
land
Charles
what's scarcity
resources short in supply
Charles
hello
scarcity is excess against human wants.
Kennedy
scarcity is limit in supply relating to demand
Akwosih
students
Hamdu
shortge of resources .imbalance of wants to resources .
Hamdu
hlo
Yahya
limitation of supply in relation to their demand for commodity
Prince
what are the two types of economic theory's?
i thick it is microeconomic theory and macroeconomic theory. or it can be normative and positive economic theories.
Deep
yes^
Nguyen
with diagrams show thé change in prices in thé different time period that can result in an increase in demande
define momentary period
Fankam
What is a monopsony?
monopsony is a situation where only one buyer is available in the market
The
And with many sellers?
Allan
oligopsony
The
to be more specific, oligopsony is a situation with many sellers but few buyers
The
Thank you
Allan
economic is tha process of banking
Pls can u explain it into details
Praise
Cause I don't understand what you are saying
Praise
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