# 6.1 Exponential functions

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In this section, you will:
• Evaluate exponential functions.
• Find the equation of an exponential function.
• Use compound interest formulas.
• Evaluate exponential functions with base e.

India is the second most populous country in the world with a population of about $\text{\hspace{0.17em}}1.25\text{\hspace{0.17em}}$ billion people in 2013. The population is growing at a rate of about $\text{\hspace{0.17em}}1.2%\text{\hspace{0.17em}}$ each year http://www.worldometers.info/world-population/. Accessed February 24, 2014. . If this rate continues, the population of India will exceed China’s population by the year $\text{\hspace{0.17em}}2031.$ When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions , which model this kind of rapid growth.

## Identifying exponential functions

When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation $\text{\hspace{0.17em}}f\left(x\right)=3x+4,$ the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.

## Defining an exponential function

A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2050.

What exactly does it mean to grow exponentially ? What does the word double have in common with percent increase ? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.

• Percent change refers to a change based on a percent of the original amount.
• Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
• Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.

For us to gain a clear understanding of exponential growth    , let us contrast exponential growth with linear growth . We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See [link] .

$x$ $f\left(x\right)={2}^{x}$ $g\left(x\right)=2x$
0 1 0
1 2 2
2 4 4
3 8 6
4 16 8
5 32 10
6 64 12

From [link] we can infer that for these two functions, exponential growth dwarfs linear growth.

• Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
• Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.

#### Questions & Answers

a diagram of an adult mosquito
mubarak Reply
what are white blood cells
Mlungisi Reply
white blood cell is part of the immune system. that help fight the infection.
MG
what about tissue celss
Mlungisi
Cells with a similar function, form a tissue. For example the nervous tissue is composed by cells:neurons and glia cells. Muscle tissue, is composed by different cells.
Anastasiya
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Isala
on what?
Anastasiya
is air homogenous or hetrogenous
damiane Reply
homogenous
Kevin
why saying homogenous?
Isala
explain if oxygen is necessary for photosynthesis
Allice Reply
explain if oxygen is necessary for photosynthesis
Allice Reply
Yes, the plant does need oxygen. The plant uses oxygen, water, light, and produced food. The plant use process called photosynthesis.
MG
By using the energy of sunlight, plants convert carbon dioxide and water into carbohydrates and oxygen by photosynthesis. This happens during the day and sunlight is needed.
NOBLE
no. it s a product of the process
Anastasiya
yet still is it needed?
NOBLE
no. The reaction is: 6CO2+6H20+ solar energy =C6H12O6(glucose)+602. The plant requires Carbon dioxyde, light, and water Only, and produces glucose and oxygen( which is a waste).
Anastasiya
what was the question
NOBLE Reply
joining
Godfrey
the specific one
NOBLE
the study of non and living organism is called.
Godfrey
Is call biology
Alohan
yeah
NOBLE
yes
Usher
what Is ecology
Musonda Reply
what is a cell
Emmanuel Reply
A cell is a basic structure and functional unit of life
Ndongya
what is biolgy
Hawwi Reply
is the study of living and non living organisms
Ahmed
may u draw the female organ
MARTIN Reply
i dont understand
Asal
:/
Asal
me too
DAVID
anabolism and catabolism
Sani Reply
Anabolism refers to the process in methabolism in which complex molecules are formed "built" and requires energy to happen. Catabolism is the opposite process: complex molecules are deconstructed releasing energy, such as during glicolysis.
Anastasiya
Explain briefly independent assortment gene .
Otu Reply
hi
Amargo
hi I'm Anatalia
Joy
what do you mean by pituitary gland
Digambar
draw and lable the cell
Ameh Reply
why homones are destroyed soon after completing their role
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