June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function calculates an exponential decrease or increase in a certain base. For example, let us suppose a country's population doubles annually. This population growth can be portrayed as an exponential function.

Exponential functions have many real-world use cases. Expressed mathematically, an exponential function is displayed as f(x) = b^x.

Here we discuss the fundamentals of an exponential function along with important examples.

What is the formula for an Exponential Function?

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

b is the base, and x is the exponent or power.
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 the event where b is greater than 0 and does not equal 1, x will be a real number.

How do you chart Exponential Functions?

To graph an exponential function, we need to locate the dots where the function intersects the axes. These are referred to as the x and y-intercepts.

Since the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To discover the y-coordinates, its essential to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

In following this approach, we get the range values and the domain for the function. Once we have the worth, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

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

The line passes 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 increasing
The graph is level and constant
As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
As x approaches positive infinity, the graph increases without bound.

In cases where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following attributes:

The graph passes the point (0,1)
The range is larger than 0
The domain is all real numbers
The graph is declining
The graph is a curved line
As x nears positive infinity, the line in the graph is asymptotic to the x-axis.
As x advances toward negative infinity, the line approaches without bound
The graph is flat
The graph is constant

Rules

There are several vital rules to bear in mind when engaging with exponential functions.

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

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

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

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

Rule 4: An exponential function with a base of 1 is consistently equal to 1.

For example, 1^x = 1 no matter what the rate of x is.

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

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

Examples

Exponential functions are usually used to indicate exponential growth. As the variable grows, the value of the function increases faster and faster.

Example 1

Let’s observe the example of the growth of bacteria. Let’s say we have a group of bacteria that duplicates every hour, then at the close of the first hour, we will have 2 times as many bacteria.

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

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

This rate of growth can be portrayed using 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

Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive substance that decays at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.

After two hours, we will have 1/4 as much substance (1/2 x 1/2).

After three hours, we will have one-eighth 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 calculated in hours.

As demonstrated, both of these samples follow a similar pattern, which is the reason they can be represented using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable while the base continues to be constant. This indicates that any exponential growth or decline where the base is different is not an exponential function.

For example, in the case of compound interest, the interest rate stays the same while the base varies in regular time periods.

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's review the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As you can see, the values of y increase very rapidly as x rises. Consider we were to draw this exponential function graph on a coordinate plane, it would look like this:

As you can see, the graph is a curved line that rises from left to right and gets steeper as it goes.

Example 2

Plot the following exponential function:

y = 1/2^x

To begin, let's create a table of values.

As shown, the values of y decrease very rapidly as x increases. The reason is because 1/2 is less than 1.

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

This is a decay function. As shown, the graph is a curved line that decreases from right to left and gets flatter as it continues.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions exhibit particular features where the derivative of the function is the function itself.

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

Exponential Series

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

Source

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