June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function measures an exponential decrease or increase in a certain base. For instance, let's say a country's population doubles every year. This population growth can be depicted in the form of an exponential function.

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

Today we will learn the essentials of an exponential function in conjunction with important examples.

What’s the equation 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 a constant, 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 a situation where b is higher than 0 and unequal to 1, x will be a real number.

How do you chart Exponential Functions?

To chart an exponential function, we need to locate the dots where the function intersects the axes. This is referred to as the x and y-intercepts.

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

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

According to this method, we determine the domain and the range values for the function. After having the values, we need to draw 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 more than 1, the graph is going to have the below characteristics:

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 rising
The graph is flat and constant
As x nears negative infinity, the graph is asymptomatic regarding the x-axis
As x nears positive infinity, the graph rises without bound.

In events where the bases are fractions or decimals between 0 and 1, an exponential function displays the following properties:

The graph crosses the point (0,1)
The range is greater than 0
The domain is entirely real numbers
The graph is decreasing
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 basic rules to bear in mind when working with exponential functions.

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

For example, if we have to multiply two exponential functions that posses 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, deduct the exponents.

For instance, if we have to divide two exponential functions that posses 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, we are able to compose it as (4^x)^3 = 4^(3x).

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

For example, 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 regardless of what 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 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 cluster of bacteria that duplicates every hour, then at the close of the first hour, we will have double as many bacteria.

At the end of the second hour, 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 displayed utilizing 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

Also, exponential functions can illustrate exponential decay. If we have a dangerous substance 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 material.

At the end of hour two, we will have 1/4 as much substance (1/2 x 1/2).

After hour three, we will have an eighth as much substance (1/2 x 1/2 x 1/2).

This can be displayed using an exponential equation as below:

f(t) = 1/2^t

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

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

As a matter of fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base continues to be fixed. This means that any exponential growth or decline where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate stays the same while the base varies in ordinary intervals 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 input different values for x and then calculate the corresponding values for y.

Let us check out the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

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

As seen above, the graph is a curved line that rises 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 shown, the values of y decrease very swiftly as x surges. 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 you can see, 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 particular properties by which 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 terminology are the powers of an independent variable digit. The general form of an exponential series is:

Source

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