June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or rise in a specific base. Take this, for example, let us assume a country's population doubles every year. This population growth can be depicted in the form of an exponential function.

Exponential functions have multiple real-world use cases. In mathematical terms, an exponential function is shown as f(x) = b^x.

Today we will review the fundamentals of an exponential function in conjunction with appropriate examples.

What is the formula for an Exponential Function?

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

b is the base, and x is the exponent or power.
b is fixed, and x is a variable

For example, if b = 2, we then 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 higher than 0 and does not equal 1, x will be a real number.

How do you graph Exponential Functions?

To chart an exponential function, we need to find the points where the function intersects the axes. This is called the x and y-intercepts.

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

To locate the y-coordinates, we need to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

By following this method, we achieve 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 identical qualities. When the base of an exponential function is more than 1, the graph would have the following properties:

The line intersects 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 constant
As x nears negative infinity, the graph is asymptomatic concerning the x-axis
As x approaches positive infinity, the graph increases without bound.

In situations 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 greater than 0
The domain is entirely real numbers
The graph is descending
The graph is a curved line
As x approaches 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 level
The graph is continuous

Rules

There are some vital rules to remember when dealing with exponential functions.

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

For example, if we need to multiply two exponential functions that have a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).

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

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

Rule 3: To increase 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, we are able to compose it as (4^x)^3 = 4^(3x).

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

For example, 1^x = 1 regardless of what the rate of x is.

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

For example, 0^x = 0 no matter what the value of x is.

Examples

Exponential functions are commonly leveraged to denote exponential growth. As the variable increases, the value of the function grows at a ever-increasing pace.

Example 1

Let's look at the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that doubles hourly, then at the close of the first hour, we will have 2 times as many bacteria.

At the end of hour two, we will have quadruple 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 displayed an exponential function as follows:

f(t) = 2^t

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

Example 2

Similarly, exponential functions can portray exponential decay. Let’s say we had a dangerous material that degenerates at a rate of half its amount every hour, then at the end of the first hour, we will have half as much substance.

After the second hour, we will have 1/4 as much material (1/2 x 1/2).

After the third hour, we will have an eighth as much material (1/2 x 1/2 x 1/2).

This can be represented using an exponential equation as below:

f(t) = 1/2^t

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

As demonstrated, both of these examples follow a similar pattern, which is the reason they are able to be depicted using exponential functions.

As a matter of 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 the same. This indicates that any exponential growth or decomposition where the base is different is not an exponential function.

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

Solution

An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we have to enter different values for x and calculate the corresponding values for y.

Let's look at the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As shown, the worth of y grow very fast as x grows. Imagine 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 goes up from left to right and gets steeper as it continues.

Example 2

Chart 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 quickly as x surges. This is because 1/2 is less than 1.

Let’s say we were to graph the x-values and y-values on a coordinate plane, it is going to look like what you see below:

This is a decay function. As you can see, the graph is a curved line that descends from right to left and gets smoother as it continues.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions display special characteristics 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 number. The common form of an exponential series is:

Source

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