June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or rise in a particular base. For instance, let us suppose a country's population doubles yearly. This population growth can be represented as an exponential function.

Exponential functions have many real-world use cases. Mathematically speaking, an exponential function is written as f(x) = b^x.

Here we will learn the basics of an exponential function in conjunction with relevant examples.

What is the equation for an Exponential Function?

The common equation 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 is a variable

For example, 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 larger 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 have to locate the dots where the function crosses the axes. These are called the x and y-intercepts.

As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.

To find the y-coordinates, one must to set the worth for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

In following this method, we determine the range values and the domain for the function. Once we determine the worth, we need to chart 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 greater than 1, the graph would have the below characteristics:

The line intersects the point (0,1)
The domain is all positive real numbers
The range is greater 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 towards the x-axis
As x advances toward positive infinity, the graph rises without bound.

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

The graph crosses the point (0,1)
The range is larger than 0
The domain is all real numbers
The graph is decreasing
The graph is a curved line
As x nears positive infinity, the line in the graph is asymptotic to the x-axis.
As x gets closer to negative infinity, the line approaches without bound
The graph is level
The graph is continuous

Rules

There are some basic rules to bear in mind when engaging with exponential functions.

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

For instance, if we have 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 an equivalent base, deduct the exponents.

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

Rule 4: An exponential function that has a base of 1 is forever equal to 1.

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

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

For instance, 0^x = 0 regardless of what the value of x is.

Examples

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

Example 1

Let’s observe the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that duplicates hourly, 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 8x as many bacteria (2 x 2 x 2).

This rate of growth can be portrayed utilizing 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 portray exponential decay. If we have a radioactive material that decays at a rate of half its quantity every hour, then at the end of one hour, we will have half as much material.

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

At the end of three hours, 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 follows:

f(t) = 1/2^t

where f(t) is the volume of substance at time t and t is calculated in hours.

As you can see, both of these examples use a comparable pattern, which is why they can be depicted using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base remains constant. Therefore any exponential growth or decline where the base changes is not an exponential function.

For example, in the scenario of compound interest, the interest rate remains the same while the base varies in normal amounts of time.

Solution

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

Let us look at the following example.

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 worth of y grow very rapidly as x rises. If we were to draw 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 and gets steeper as it persists.

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 rapidly as x increases. This is because 1/2 is less than 1.

If we were to chart the x-values and y-values on a coordinate plane, it is going to look like this:

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

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 unique 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 number. The general form of an exponential series is:

Source

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