Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or rise in a particular base. For example, let us suppose a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.

Exponential functions have numerous real-world applications. In mathematical terms, an exponential function is written as f(x) = b^x.

In this piece, we will learn the basics of an exponential function in conjunction with relevant examples.

What’s the formula for an Exponential Function?

The general formula for an exponential function is f(x) = b^x, where:

1. b is the base, and x is the exponent or power.

2. b is a constant, and x varies

As an illustration, if b = 2, then we get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.

In a situation where b is greater than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To chart an exponential function, we need to locate the spots where the function crosses the axes. These are referred to as the x and y-intercepts.

Considering the fact that the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.

To find the y-coordinates, we need to set the value for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.

According to this approach, we achieve the domain and the range values for the function. Once we have the rate, we need to graph them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical characteristics. When the base of an exponential function is more than 1, the graph will have the below qualities:

• The line crosses the point (0,1)

• The domain is all positive real numbers

• The range is larger than 0

• The graph is a curved line

• The graph is on an incline

• The graph is smooth and continuous

• As x nears negative infinity, the graph is asymptomatic regarding the x-axis

• As x nears positive infinity, the graph grows without bound.

In instances where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following properties:

• The graph passes the point (0,1)

• The range is more than 0

• The domain is all real numbers

• The graph is descending

• The graph is a curved line

• As x nears positive infinity, the line within graph is asymptotic to the x-axis.

• As x advances toward negative infinity, the line approaches without bound

• The graph is level

• The graph is continuous

Rules

There are a few essential rules to remember when working with exponential functions.

Rule 1: Multiply exponential functions with the same base, add the exponents.

For instance, if we need to multiply two exponential functions with a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with an identical base, subtract the exponents.

For example, if we have to divide two exponential functions that have a base of 3, we can compose it as 3^x / 3^y = 3^(x-y).

Rule 3: To raise an exponential function to a power, multiply the exponents.

For instance, if we have to increase an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is always equivalent to 1.

For instance, 1^x = 1 no matter what the worth of x is.

Rule 5: An exponential function with a base of 0 is always equal to 0.

For example, 0^x = 0 despite whatever the value of x is.

Examples

Exponential functions are commonly leveraged to signify exponential growth. As the variable grows, the value of the function grows quicker and quicker.

Example 1

Let’s examine the example of the growth of bacteria. Let us suppose that we have a culture of bacteria that multiples by two each hour, then at the end of hour one, we will have 2 times as many bacteria.

At the end of hour two, we will have 4x as many bacteria (2 x 2).

At the end of the third hour, we will have 8 times as many bacteria (2 x 2 x 2).

This rate of growth can be portrayed an exponential function as follows:

f(t) = 2^t

where f(t) is the number of bacteria at time t and t is measured hourly.

Example 2

Similarly, exponential functions can illustrate exponential decay. If we have a radioactive substance that decays at a rate of half its volume every hour, then at the end of hour one, we will have half as much substance.

At the end of two hours, we will have a quarter as much material (1/2 x 1/2).

At the end of the third hour, we will have 1/8 as much material (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the quantity of material at time t and t is measured in hours.

As shown, both of these illustrations follow a comparable pattern, which is why they are able to be shown using exponential functions.

In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base continues to be fixed. This indicates that any exponential growth or decay where the base changes is not an exponential function.

For example, in the matter of compound interest, the interest rate stays the same while the base changes in normal amounts of time.

Solution

An exponential function can be graphed using a table of values. To get the graph of an exponential function, we need to plug in different values for x and then calculate the matching values for y.

Let us review the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

To start, let's make a table of values.

As you can see, the values of y grow very quickly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like this:

As seen above, the graph is a curved line that rises from left to right ,getting steeper as it continues.

Example 2

Draw the following exponential function:

y = 1/2^x

First, let's create a table of values.

As you can see, the values of y decrease very rapidly as x rises. This is because 1/2 is less than 1.

Let’s say we were to plot the x-values and y-values on a coordinate plane, it would look like the following:

The above is a decay function. As shown, the graph is a curved line that decreases from right to left and gets smoother as it goes.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be written as f(ax)/dx = ax. All derivatives of exponential functions present special characteristics where the derivative of the function is the function itself.

The above can be written as following: f'x = a^x = f(x).

Exponential Series

The exponential series is a power series whose terms are the powers of an independent variable digit. The general form of an exponential series is:

Source

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