## Exponential Functions

An exponential function is a mathematical function, which is used in many real-world situations. It is mainly used to find the exponential decay or exponential growth or to compute investments, model populations and so on. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives, exponential series and examples.

## What is Exponential Function?

An exponential function is a Mathematical function in form f (x) = a^{x}, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.

## Exponential Function Formula

An exponential function is defined by the formula f(x) = a^{x}, where the input variable x occurs as an exponent. The exponential curve depends on the exponential function and it depends on the value of the x.

The exponential function is an important mathematical function which is of the form

**f(x) = a ^{x}**

Where a>0 and a is not equal to 1.

x is any real number.

If the variable is negative, the function is undefined for -1 < x < 1.

Here,

“x” is a variable

“a” is a constant, which is the base of the function.

An exponential curve grows, or decay depends on the exponential function. Any quantity that grows or decays by a fixed per cent at regular intervals should possess either exponential growth or exponential decay.

**Exponential Growth**

In Exponential Growth, the quantity increases very slowly at first, and then rapidly. The rate of change increases over time. The rate of growth becomes faster as time passes. The rapid growth meant to be an “exponential increase”. The formula to define the exponential growth is:

**y = a ( 1+ r ) ^{x}**

Where r is the growth percentage.

**Exponential Decay**

In Exponential Decay, the quantity decreases very rapidly at first, and then slowly. The rate of change decreases over time. The rate of change becomes slower as time passes. The rapid growth meant to be an “exponential decrease”. The formula to define the exponential growth is:

**y = a ( 1- r ) ^{x}**

## Exponential Function Graph

The following figure represents the graph of exponents of x. It can be seen that as the exponent increases, the curves get steeper and the rate of growth increases respectively. Thus, for x > 1, the value of y = f_{n}(x) increases for increasing values of (n).

From the above, it can be seen that the nature of polynomial functions is dependent on its degree. Higher the degree of any polynomial function, then higher is its growth. A function which grows faster than a polynomial function is y = f(x) = a^{x}, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of f_{n}(x).

Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = a^{x}. The domain of exponential function will be the set of entire real numbers R and the range are said to be the set of all the positive real numbers.

It must be noted that exponential function is increasing and the point (0, 1) always lies on the graph of an exponential function. Also, it is very close to zero if the value of x is mostly negative.

Exponential function having base 10 is known as a common exponential function. Consider the following series:

The value of this series lies between 2 & 3. It is represented by e. Keeping e as base the function, we get y = e^{x}, which is a very important function in mathematics known as a natural exponential function.

For a > 1, the logarithm of b to base a is x if a^{x} = b. Thus, log_{a} b = x if a^{x} = b. This function is known as logarithmic function.

For base a = 10, this function is known as common logarithm and for the base a = e, it is known as natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which have a base a>1.

- The domain of log function consists of positive real numbers only, as we cannot interpret the meaning of log functions for negative values.
- For the log function, though the domain is only the set of positive real numbers, the range is set of all real values, i.e. R
- When we plot the graph of log functions and move from left to right, the functions show increasing behaviour.
- The graph of log function never cuts x-axis or y-axis, though it seems to tend towards them.

- Log
_{a}p = α, log_{b}p = β and log_{b}a = µ, then a^{α }= p, b^{β }= p and b^{µ }= a - Log
_{b}pq = Log_{b}p + Log_{b}q - Log
_{b}p^{y }= ylog_{b}p - Log
_{b }(p/q) = log_{b}p – log_{b}q

## Exponential Function Derivative

Let us now focus on the derivative of exponential functions.

**The derivative of e ^{x }with respect to x is e^{x}, i.e. d(e^{x})/dx = e^{x}**

It is noted that the exponential function f(x) =e^{x }has a special property. It means that the derivative of the function is the function itself.

(i.e) f ‘(x) = e^{x }= f(x)

## Exponential Series

The exponential series are given below.

## Exponential Function Properties

The exponential graph of a function represents the exponential function properties.

Let us consider the exponential function, y=2^{x}

The graph of function y=2^{x} is shown below. First, the property of the exponential function graph when the base is greater than 1.

**Exponential Function Graph for y=2 ^{x}**

The graph passes through the point (0,1).

- The domain is all real numbers
- The range is y>0
- The graph is increasing
- The graph is asymptotic to the x-axis as x approaches negative infinity
- The graph increases without bound as x approaches positive infinity
- The graph is continuous
- The graph is smooth

**Exponential Function Graph y=2 ^{-x} **

The graph of function y=2^{-x} is shown above. The properties of the exponential function and its graph when the base is between 0 and 1 are given.

- The line passes through the point (0,1)
- The domain includes all real numbers
- The range is of y>0
- It forms a decreasing graph
- The line in the graph above is asymptotic to the x-axis as x approaches positive infinity
- The line increases without bound as x approaches negative infinity
- It is a continuous graph
- It forms a smooth graph

### Exponential Function Rules

Some important exponential rules are given below:

If a>0, and b>0, the following hold true for all the real numbers x and y:

- a
^{x}a^{y}= a^{x+y} - a
^{x}/a^{y}= a^{x-y} - (a
^{x})^{y}= a^{xy} - a
^{x}b^{x}=(ab)^{x} - (a/b)
^{x}= a^{x}/b^{x} - a
^{0}=1 - a
^{-x}= 1/ a^{x}

### Exponential Functions Examples

The examples of exponential functions are:

- f(x) = 2
^{x} - f(x) = 1/ 2
^{x}= 2^{-x} - f(x) = 2
^{x+3} - f(x) = 0.5
^{x}

### Solved problem

**Question:**

Simplify the exponential equation 2^{x}-2^{x+1}

**Solution:**

Given exponential equation: 2^{x}-2^{x+1}

By using the property: a^{x }a^{y }= a^{x+y}

Hence, 2^{x+1 }can be written as 2^{x}. 2

^{ }Thus the given equation is written as:

2^{x}-2^{x+1 }=2^{x}-2^{x}. 2

Now, factor out the term 2^{x}

2^{x}-2^{x+1 }=2^{x}-2^{x}. 2 = 2^{x}(1-2)

2^{x}-2^{x+1 }= 2^{x}(-1)

2^{x}-2^{x+1 }= – 2^{x}

Therefore, the simplification of the given expoential equation 2^{x}-2^{x+1} is – 2^{x}.

## Exponential Function

The exponential function is a type of mathematical function which are helpful in finding the growth or decay of population, money, price, etc that are growing or decay exponentially. Jonathan was reading a news article on the latest research made on bacterial growth. He read that an experiment was conducted with one bacterium. After the first hour, the bacterium doubled itself and was two in number. After the second hour, the number was four. At every hour the number of bacteria was increasing. He was thinking what would be the number of bacteria after 100 hours if this pattern continues. When he asked his teacher about the same the answer he got was the concept of an exponential function.

Let us learn more about exponential function along with its definition, equation, graphs, exponential growth, exponential decay, etc.

## What is Exponential Function?

Exponential function, as its name suggests, involves exponents. But note that, an exponential function has a constant as its base and a variable as its exponent but not the other way round (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function). An exponential function can be in one of the following forms.

### Exponential Function Definition

In mathematics, an exponential function is a function of form f (x) = a^{x}, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0.

### Exponential Function Examples

Here are some examples of exponential function.

- f(x) = 2
^{x} - f(x) = (1/2)
^{x} - f(x) = 3e
^{2x} - f(x) = 4 (3)
^{-0.5x}

## Exponential Function Formula

A basic exponential function, from its definition, is of the form f(x) = b^{x}, where ‘b’ is a constant and ‘x’ is a variable. One of the popular exponential functions is f(x) = e^{x}, where ‘e’ is “Euler’s number” and e = 2.718….If we extend the possibilities of different exponential functions, an exponential function may involve a constant as a multiple of the variable in its power. i.e., an exponential function can also be of the form f(x) = e^{kx}. Further, it can also be of the form f(x) = p e^{kx}, where ‘p’ is a constant. Thus, an exponential function can be in one of the following forms.

- f(x) = b
^{x} - f(x) = ab
^{x} - f(x) = ab
^{cx} - f(x) = e
^{x} - f(x) = e
^{kx} - f(x) = p e
^{kx}

Here, apart from ‘x’ all other letters are constants, ‘x’ is a variable, and f(x) is an exponential function in terms of x. Also, note that the base in each exponential function must be a positive number. i.e., in the above functions, b > 0 and e > 0. Also, b should not be equal to 1 (if b = 1, then the function f(x) = b^{x} becomes f(x) = 1 and in this case, the function is linear but NOT exponential).

The exponential function arises whenever a quantity’s value increases in exponential growth and decreases in exponential decay. We can see more differences between exponential growth and decay along with their formulas in the following table.

Exponential Growth | Exponential Decay |
---|---|

In exponential growth, a quantity slowly increases in the beginning and then it increases rapidly. | In exponential decay, a quantity decreases very rapidly in the beginning, and then it decreases slowly. |

The exponential growth formulas are used to model population growth, to model compound interest, to find doubling time, etc | The exponential decay is helpful to model population decay, to find half-life, etc. |

The graph of the function in exponential growth is increasing. | The graph of the function in exponential growth is decreasing. |

In exponential growth, the function can be of the form: - f(x) = ab
^{x}, where b > 1. - f(x) = a (1 + r)
^{x} - P = P0 e
^{k t}
Here, b = 1 + r ≈ e |
In exponential decay, the function can be of the form: - f(x) = ab
^{x}, where 0 < b < 1. - f(x) = a (1 – r)
^{x} - P = P0 e
^{-k t}
Here, b = 1 – r ≈ e |

In the above formulas,

- a (or) P00 = Initial value
- r = Rate of growth
- k = constant of proportionality
- x (or) t = time (time can be in years, days, (or) months. Whatever we are using should be consistent throughout the problem).

## Exponential Series

The real exponential function can be commonly defined by the following power series,

Exponential Function Rules

The rules of exponential function are as same as that of rules of exponents. Here are some rules of exponents.

- Law of Zero Exponent: a
^{0}= 1 - Law of Product: a
^{m}× a^{n}= a^{m+n} - Law of Quotient: a
^{m}/a^{n}= a^{m-n} - Law of Power of a Power: (a
^{m})^{n}= a^{mn} - Law of Power of a Product: (ab)
^{m}= a^{m}b^{m} - Law of Power of a Quotient: (a/b)
^{m}= a^{m}/b^{m} - Law of Negative Exponent: a
^{-m}= 1/a^{m}

Apart from these, we sometimes need to use the conversion formula of logarithmic form to exponential form which is:

- b
^{x }= a ⇔ logbb a = x

### Equality Property of Exponential Function

According to the equality property of exponential function, if two exponential functions of the same bases are the same, then their exponents are also the same. i.e.,

b^{x1} = b^{x2 }⇔ x1 = x2

## Exponential Function Graph

We can understand the process of graphing exponential function by taking some examples. Let us graph two functions f(x) = 2^{x} and g(x) = (1/2)^{x}. To graph each of these functions, we will construct a table of values with some random values of x, plot the points on the graph, connect them by a curve, and extend the curve on both ends.

Here is the table of values that are used to graph the exponential function f(x) = 2^{x}.

Here is the table of values that are used to graph the exponential function g(x) = (1/2)^{x}.

**Note: **From the above two graphs, we can see that f(x) = 2^{x} is increasing whereas g(x) = (1/2)^{x} is decreasing. Thus, the graph of exponential function f(x) = b^{x}.

- increases when b > 1
- decreases when 0 < b < 1

## Domain and Range of Exponential Function

We know that the domain of a function y = f(x) is the set of all x-values (inputs) where it can be computed and the range is the set of all y-values (outputs) of the function. From the graphs of f(x) = 2^{x} and g(x) = (1/2)^{x} in the previous section, we can see that an exponential function can be computed at all values of x. Thus, the domain of an exponential function is the set of all real numbers (or) (-∞, ∞). The range of an exponential function can be determined by the horizontal asymptote of the graph, say, y = d, and by seeing whether the graph is above y = d or below y = d. Thus, for an exponential function f(x) = ab^{x},

- Domain is the set of all real numbers (or) (-∞, ∞).
- Range is f(x) > d if a > 0 and f(x) < d if a < 0.

To understand this, you can see the example below.

## Exponential Function Derivative

Here are the formulas from differentiation that are used to find the derivative of exponential function.

- d/dx (e
^{x}) = e^{x} - d/dx (a
^{x}) = a^{x}· ln a.

## Integration of Exponential Function

Here are the formulas from integration that are used to find the integral of exponential function.

- ∫ e
^{x}dx = e^{x}+ C - ∫ a
^{x}dx = a^{x}/ (ln a) + C

## Examples on Exponential Function

**Example 1:** In 2010, there were 100,000 citizens in a town. If the population increases by 8% every year, then how many citizens will there be in 10 years? Round your answer to the nearest integer.

**Solution:**

The initial population is, a = 100,000.

The rate of growth is, r = 8% = 0.08.

The time is, x = 10 years.

Using the exponential growth formula,

f(x) = a (1 + r)^{x}

f(x) = 100000(1 + 0.08)^{10}

f(x) ≈ 215,892 (rounded to the nearest integer)

**Answer:** Therefore, the number of citizens in 10 years will be 215,892.

**Example 2:** The half-life of carbon-14 is 5,730 years. If there were 1000 grams of carbon initially, then what is the amount of carbon left after 2000 years? Round your answer to the nearest integer.

**Solution:**

Using the given data, we can say that carbon-14 is decaying and hence we use the formula of exponential decay.

P = P0 e^{– k t} … (1),

Here, P0 = initial amount of carbon = 1000 grams.

It is given that the half-life of carbon-14 is 5,730 years. It means

P = P0 / 2 = 1000 / 2 = 500 grams.

Substitute all these values in (1),

500 = 1000 e^{– k (5730)}

Dividing both sides by 1000,

0.5 = e^{– k (5730)}

Taking “ln” on both sides,

ln 0.5 = -5730k

Dividing both sides by -5730,

k = ln 0.5 / (-5730) ≈ 0.00012097

We have to find the amount of carbon that is left after 2000 years. Substitute t = 2000 in (1),

P = 1000 e^{– (0.00012097) (2000)} ≈ 785 grams.

**Answer: **The amount of carbon left after 1000 years = 785 grams.

**Example 3:** Simplify the following exponential expression: 3^{x }– 3^{x+2}

**Solution:**

Given exponential equation: 3^{x }– 3^{x+2}

By using the property: a^{x }a^{y }= a^{x+y}

Hence, 3^{x+2 }can be written as 3^{x}.3^{2}

Thus the given equation is written as:

3^{x }– 3^{x+2 }= 3^{x }– 3^{x}·9

Now, factor out the term 3^{x}

3^{x }– 3^{x+2 }= 3^{x }– 3^{x}·9 = 3^{x}(1 – 9)

3^{x }– 3^{x+2 }= 3^{x}(-8)

3^{x }– 3^{x+2 }= -8(3^{x})

Therefore, the simplification of the given expoential equation 3^{x}-3^{x+1} is -8(3^{x}).

## FAQs on Exponential Function

### What is Exponential Function?

An exponential function is a type of function in math that involves exponents. A basic exponential function is of the form f(x) = b^{x}, where b > 0 and b ≠ 1.

### What are the Formulas of Exponential Function?

The formulas of an exponential function have exponents in them. An exponential equation can be in one of the following forms.

- f(x) = ab
^{cx} - f(x) = p e
^{kx}

### What are the Rules of Exponential Function?

Since the exponential function involves exponents, the rules of exponential function are as same as the rules of exponents. They are:

- a
^{m}× a^{n}= a^{m+n} - a
^{m}/a^{n}= a^{m-n} - a
^{0}= 1 - a
^{-m}= 1/a^{m} - (a
^{m})^{n}= a^{mn} - (ab)
^{m}= a^{m}b^{m} - (a/b)
^{m}= a^{m}/b^{m}

### How to Graph Exponential Function?

To graph an exponential function y = f(x), create a table of values by taking some random numbers for x (usually we take -2, -1, 0, 1, and 2), substitute each of them in the function to find the corresponding y values. Then plot the points from the table and join them by a curve. Finally, extend the curve on both ends.

### What is the Domain of Exponential Function?

An exponential function f(x) = ab^{x} is defined for all values of x and hence its domain is the set of all real numbers, which in interval notation can be written as (-∞, ∞).

### What are the Properties of Exponential Function?

The properties of exponential function can be given as,

- a
^{m}× a^{n}= a^{m+n} - a
^{m}/a^{n}= a^{m-n} - a
^{0}= 1 - a
^{-m}= 1/a^{m} - (a
^{m})^{n}= a^{mn} - (ab)
^{m}= a^{m}b^{m} - (a/b)
^{m}= a^{m}/b^{m}

For any exponential function of the form f(x) = ab^{x}, where b > 1, the exponential graph increases while for any exponential function of the form f(x) = ab^{x}, where 0 < b < 1, the graph decreases.

### What is the Range of Exponential Function?

The range of an exponential function depends upon its horizontal asymptote and also whether the curve lies above or below the horizontal asymptote. i.e., for an exponential function f(x) = ab^{x}, the range is

- f(x) > d if a > 0 and
- f(x) < d if a < 0,

where y = d is the horizontal asymptote of the graph of the function.

### What is the Equality Property of Exponential Function?

The equality property of exponential function says if two values (outputs) of an exponential function are equal, then the corresponding inputs are also equal. i.e., b^{x1} = b^{x2 }⇔ x1 = x2.

### What is the Derivative of Exponential Function?

An exponential function may be of the form e^{x} or a^{x}. The formulas to find the derivatives of these functions are as follows:

- d/dx (e
^{x}) = e^{x} - d/dx (a
^{x}) = a^{x}· ln a.

### What is the Integral of Exponential Function?

An exponential function may be of the form e^{x} or a^{x}. The formulas to find the integrals of these functions are as follows:

- ∫ e
^{x}dx = e^{x}+ C - ∫ a
^{x}dx = a^{x}/ (ln a) + C

**Introduction to Exponential Functions**

**Learning Objective(s)**

· Graph exponential equations and functions.

· Solve applied problems using exponential problems and their graphs.

**Introduction**

In addition to linear, quadratic, rational, and radical functions, there are **exponential functions**.Exponential functions have the form *f*(*x*) = *b ^{x}*, where

*b*> 0 and

*b*≠ 1. Just as in any exponential expression,

*b*is called the

**base**and

*x*is called the

**exponent**.

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2* ^{x}* bacteria after

*x*hours. This can be written as

*f*(

*x*) = 2

*.*

^{x}Before you start, *f*(0) = 2^{0 }= 1

After 1 hour *f*(1) = 2^{1} = 2

In 2 hours *f*(2) = 2^{2} = 4

In 3 hours *f*(3) = 2^{3} = 8

and so on.

With the definition *f*(*x*) = *b ^{x}* and the restrictions that

*b*> 0 and that

*b*≠ 1, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows

*f*(

*x*) = 2

*.*

^{x}**Exponential Growth**

As you can see above, this exponential function has a graph that gets very close to the *x*-axis as the graph extends to the left (as *x* becomes more negative), but never really touches the *x*-axis. Knowing the general shape of the graphs of exponential functions is helpful for graphing specific exponential equations or functions.

Making a table of values is also helpful, because you can use the table to place the curve of the graph more accurately. One thing to remember is that if a base has a negative exponent, then take the reciprocal of the base to make the exponent positive. For example,

**Example**

Look at the table of values. Think about what happens as the *x* values increase—so do the function values (*f*(*x*) or *y*)!

Now that you have a table of values, you can use these values to help you draw both the shape and location of the function. Connect the points as best you can to make a smooth curve (not a series of straight lines). This shows that all of the points on the curve are part of this function.

This is an example of **exponential growth**. As *x* increases, *f*(*x*) “grows” more quickly. Let’s try another one.

Let’s compare the three graphs you’ve seen. The functions *f*(*x*) = 2* ^{x}*,

*f*(x) = 3

*, and*

^{x}*f*(*x*) = 4* ^{x}* are all graphed below.

Notice that a larger base makes the graph steeper. A larger base also makes the graph closer to the *y*-axis for *x* > 0 and closer to the *x*-axis for *x* < 0. All the graphs go through (0, 1)!

**Exponential Decay**

Remember that for exponential functions, *b* > 0, but *b* ≠ 1. In the examples above, *b* > 1. What happens when *b* is between 0 and 1, 0 < *b* < 1?

Notice that the shape is similar to the shape when *b* > 1, but this time the graph gets closer to the *x*-axis when *x* > 0, rather than when *x* < 0. This is **exponential decay**. Instead of the function values “growing” as *x* values increase, as they did before, the function values “decay” or decrease as *x* values increase. They get closer and closer to 0.

Incorrect. This graph is increasing, because the *f*(*x*) or *y* values increase as the *x* values increase. (Compare the values for *x* = 1 and *x* = 2.) This graph shows exponential growth, with a base greater than 1. The correct answer is Graph D.

Incorrect. This graph is decreasing, but all the function values are negative. The range for an exponential function is always positive values. The correct answer is Graph D.

Incorrect. This graph is increasing, but all the function values are negative. The correct graph must be decreasing with positive function values. The correct answer is Graph D.

Correct. All the function values are positive, and the graph is decreasing (showing exponential decay).

**Applying Exponential Functions**

Exponential functions can be used in many contexts, such as compound interest (money), population growth, and radioactive decay. In most of these, however, the function is not exactly of the form *f*(*x*) = *b ^{x}*. Often, this is adjusted by adding or multiplying constants.

For example, the compound interest formula is

, where *P* is the **principal** (the initial investment that is gathering interest) and *A* is the amount of money you would have, with interest, at the end of *t* years, using an annual interest rate of *r* (expressed as a decimal) and *m* compounding periods per year. In this case the base is the value represented by the expression

Radioactive decay is an example of exponential decay. Radioactive elements have a **half-life**. This is the amount of time it takes for half of a mass of the element to decay into another substance. For example, uranium-238 is a slowly decaying radioactive element with a half-life of about 4.47 *billion* years. That means it will take that long for 100 grams of uranium-238 to turn into 50 grams of uranium-238 (the other 50 grams will have turned into another element). That’s a long time! On the other extreme, radon-220 has a half-life of about 56 seconds. What does this mean? 100 grams of radon-220 will turn into 50 grams of radon-220 and 50 grams of something else in less than a minute!

Since the amount is halved each half-life, an exponential function can be used to describe the amount remaining over time. The formula

gives the remaining amount *R* from an initial amount *A,* where *h *is the half-life of the element and *t* is the amount of time passed (using the same time unit as the half-life).

Billy’s mother put $100 into a bank account for him when he was born. The account gained interest at a rate of 3% per year, compounded monthly. Assuming no more money was deposited and none was withdrawn, how much money will be in the account when Billy turns 18?

A) $170.24

B) $171.49

C) $8561.76

D) $20,718.34

**Answer**

A) $170.24

Incorrect. This is the amount when the annual rate is compounded annually (*m* = 1). In this case we are compounding monthly, *m* = 12. The correct answer is $171.49.

B) $171.49

Correct. Using a rate *r* of 0.03 and 12 compounding periods per year (*m* = 12), the formula gives $171.49.

C) $8561.76

Incorrect. You may have used *r* = 3 rather than *r* = 0.03 for the annual rate, then misread the result from your calculator. The correct answer is $171.49.

D) $20,718.34

Incorrect. You may have used *r* = 0.3 rather than *r* = 0.03 for the annual rate. The correct answer is $171.49.

**Summary**

Exponential functions of the form *f*(*x*) = *b ^{x}* appear in different contexts, including finance and radioactive decay. The base

*b*must be a positive number and cannot be 1. The graphs of these functions are curves that increase (from left to right) if

*b*> 1, showing exponential growth, and decrease if 0 <

*b*< 1, showing exponential decay.

#### Overview of the exponential function

The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance). To form an exponential function, we let the independent variable be the exponent. A simple example is the function

As illustrated in the above graph of f, the exponential function increases rapidly. Exponential functions are solutions to the simplest types of dynamical systems. For example, an exponential function arises in simple models of bacteria growth

An exponential function can describe growth or decay. The function g(x)=(^{1}/_{2})^{x}

is an example of exponential decay. It gets rapidly smaller as x increases, as illustrated by its graph.

In the exponential growth of f(x), the function doubles every time you add one to its input x. In the exponential decay of g(x), the function shrinks in half every time you add one to its input x. The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay.

#### Parameters of the exponential function

As with any function, the action of an exponential function f(x) can be captured by the function machine metaphor that takes inputs x and transforms them into the outputs f(x).

The function machine metaphor is useful for introducing parameters into a function. The above exponential functions f(x) and g(x) are two different functions, but they differ only by the change in the base of the exponentiation from 2 to 1/2. We could capture both functions using a single function machine but dials to represent parameters influencing how the machine works.

We could represent the base of the exponentiation by a parameter b. Then, we could write ff as a function with a single parameter (a function machine with a single dial):

f(x)=bx.

When b=2, we have our original exponential growth function f(x), and when b=^{1}/_{2}, this same ff turns into our original exponential decay function g(x). We could think of a function with a parameter as representing a whole family of functions, with one function for each value of the parameter.

We can also change the exponential function by including a constant in the exponent. For example, the function h(x)=2^{3x}

is also an exponential function. It just grows faster than f(x)=2^{x} since h(x) doubles every time you add only ^{1}/_{3} to its input x. We can introduce another parameter k into the definition of the exponential function, giving us two dials to play with. If we call this parameter kk, we can write our exponential function ff as

f(x)=b^{kx}.

You can explore the influence of both parameters b and kk in the following applet.

## Exponential Functions

Consider these two companies:

- Company A has 100 stores, and expands by opening 50 new stores a year
- Company B has 100 stores, and expands by increasing the number of stores by 50% of their total each year.

Company A is exhibiting linear growth. In linear growth, we have a constant rate of change—a constant number that the output increased for each increase in input. For company A, the number of new stores per year is the same each year.

Company B is different—we have a percent rate of change rather than a constant number of stores/year as our rate of change. To see the significance of this difference compare a 50% increase when there are 100 stores to a 50% increase when there are 1000 stores:

- 100 stores, a 50% increase is 50 stores in that year.
- 1000 stores, a 50% increase is 500 stores in that year.

Calculating the number of stores after several years, we can clearly see the difference in results.

Years | Company A | Company B |

2 | 200 | 225 |

4 | 300 | 506 |

6 | 400 | 1139 |

8 | 500 | 2563 |

10 | 600 | 5767 |

Graphs of data from A and B, with B fit to a curve.

This percent growth can be modeled with an exponential function.

## Exponential Function

An**exponential growth or decay function** is a function that grows or shrinks at a constant percent growth rate. The equation can be written in the form *f*(*x*) = *a*(1 + *r*)* ^{x}* or

*f*(

*x*) =

*a*

*b*where

^{x}*b*= 1 +

*r*.

Where

*a*is the initial or starting value of the function,*r*is the percent growth or decay rate, written as a decimal,*b*is the growth factor or growth multiplier. Since powers of negative numbers behave strangely, we limit b to positive values.

## Graphical Features of Exponential Functions

Graphically, in the function*f*(*x*) = *ab*^{x}.

*a*is the vertical intercept of the graph.*b*determines the rate at which the graph grows:- the function will increase if
*b*> 1, - the function will decrease if 0 <
*b*< 1.

- the function will increase if
- The graph will have a horizontal asymptote at
*y*= 0.

The graph will be concave up if a>0; concave down if*a*< 0. - The domain of the function is all real numbers.
- The range of the function is (0,∞) if
*a*> 0, and (−∞,0) if*a*< 0.

When sketching the graph of an exponential function, it can be helpful to remember that the graph will pass through the points (0,*a*) and (1,* ab*).

The value *b*

will determine the function’s long run behavior:

- If
*b*> 1, as*x*→ ∞,*f*(*x*) → ∞, and as*x*→ −∞,*f*(*x*) → 0. - If 0 <
*b*< 1, as*x*→ ∞,*f*(*x*) → 0, and as*x*→ –∞,*f*(*x*) → ∞.

#### Example 6

Sketch a graph of

f(x)=4(^{1}/_{3})^{x}

This graph will have a vertical intercept at (0,4), and pass through the point

(1,frac43)(1,frac43). Since *b* < 1, the graph will be decreasing towards zero. Since *a* > 0, the graph will be concave up.

We can also see from the graph the long run behavior: as*x* → ∞ , *f*(*x*) → 0, and as *x* → –∞, *f*(*x*) → ∞.

To get a better feeling for the effect of*a* and *b* on the graph, examine the sets of graphs below. The first set shows various graphs, where *a* remains the same and we only change the value for *b*. Notice that the closer the value of is to 1, the less steep the graph will be.

Changing the value of*b*.

In the next set of graphs,*a *is altered and our value for *b *remains the same.

Changing the value of *a*.

Notice that changing the value for a changes the vertical intercept. Since *a *is multiplying the*b ^{x}* term,

*a*acts as a vertical stretch factor, not as a shift. Notice also that the long run behavior for all of these functions is the same because the growth factor did not change and none of these values introduced a vertical flip.

### Example 7

Match each equation with its graph.

- f(x)=2(1.3)
^{x} - g(x)=2(1.8)
^{x} - h(x)=4(1.3)
^{x} - k(x)=4(0.7)
^{x}

The graph of*k*(*x*) is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of *h*(*x*) can be identified as the only growing exponential function with a vertical intercept at (0,4). The graphs of *f*(*x*) and *g*(*x*) both have a vertical intercept at (0,2), but since *g*(*x*) has a larger growth factor, we can identify it as the graph increasing faster.

## Applications of Exponential Functions

**Example 1**

India’s population was 1.14 billion in the year 2008 and is growing by about 1.34% each year. Write an exponential function for India’s population, and use it to predict the population in 2020.

Using 2008 as our starting time (

), our initial population will be 1.14 billion. Since the percent growth rate was 1.34%, our value for is 0.0134.

Using the basic formula for exponential growth

we can write the formula,*f*(*t*) = 1.14(1 + 0.0134)^{t}

To estimate the population in 2020, we evaluate the function at

, since 2020 is 12 years after 2008:

#### Example 2

A certificate of deposit (CD) is a type of savings account offered by banks, typically offering a higher interest rate in return for a fixed length of time you will leave your money invested. If a bank offers a 24 month CD with an annual interest rate of 1.2% compounded monthly, how much will a $1000 investment grow to over those 24 months?

First, we must notice that the interest rate is an annual rate, but is compounded monthly, meaning interest is calculated and added to the account monthly. To find the monthly interest rate, we divide the annual rate of 1.2% by 12 since there are 12 months in a year: 1.2%/12 = 0.1%. Each month we will earn 0.1% interest. From this, we can set up an exponential function, with our initial amount of $1000 and a growth rate of

, and our input measured in months:

After 24 months, the account will have grown to

**Example 3**

Bismuth-210 is an isotope that radioactively decays by about 13% each day, meaning 13% of the remaining Bismuth-210 transforms into another atom (polonium-210 in this case) each day. If you begin with 100 mg of Bismuth-210, how much remains after one week?

With radioactive decay, instead of the quantity increasing at a percent rate, the quantity is decreasing at a percent rate. Our initial quantity is

mg, and our growth rate will be negative 13%, since we are decreasing: . This gives the equation

This can also be explained by recognizing that if 13% decays, then 87% remains.

After one week, 7 days, the quantity remaining would be

mg of Bismuth-210 remains.

## Euler’s Number: e

Because e is often used as the base of an exponential, most scientific and graphing calculators have a button that can calculate powers of e, usually labeled exp(x). Some computer software instead defines a function exp(x), where exp(x) = . Since calculus studies continuous change, we will almost always use the -based form of exponential equations in this course.

## Continuous Growth Formula

**Continuous growth** can be calculated using the formula where

- is the starting amount,
- is the continuous growth rate.

#### Example 4

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100mg of Radon-222 decay to in 3 days?

Since we are given a continuous decay rate, we use the continuous growth formula. Since the substance is decaying, we know the growth rate will be negative:

, mg of Radon-222 will remain.

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

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