June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or increase in a specific base. Take this, for example, let us suppose a country's population doubles annually. This population growth can be represented as an exponential function.

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

Here we will review the basics of an exponential function coupled with important examples.

What is the formula for an Exponential Function?

The generic 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

For instance, 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 larger than 0 and does not equal 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we must discover the dots where the function intersects the axes. This is 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, we need to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

According to this technique, we determine the domain and the range values for the function. After having 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 more than 1, the graph would have the below properties:

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 smooth and constant
As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
As x advances toward positive infinity, the graph grows without bound.

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

The graph passes 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 nears positive infinity, the line within 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 constant

Rules

There are some basic rules to remember when engaging 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 with 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 identical base, subtract the exponents.

For instance, if we have 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 raise an exponential function with a base of 4 to the third power, we are able to note it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is consistently 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 example, 0^x = 0 no matter what the value of x is.

Examples

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

Example 1

Let’s examine the example of the growing of bacteria. Let us suppose that we have a group of bacteria that multiples by two every hour, then at the end of the first hour, we will have twice as many bacteria.

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

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

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

f(t) = 2^t

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

Example 2

Also, exponential functions can portray exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its volume every hour, then at the end of hour one, we will have half as much material.

At the end of hour two, we will have one-fourth as much material (1/2 x 1/2).

After hour three, we will have 1/8 as much substance (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 quantity of substance at time t and t is measured in hours.

As demonstrated, both of these examples pursue a similar pattern, which is the reason they can be shown using exponential functions.

As a matter of fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains the same. Therefore any exponential growth or decomposition where the base changes is not an exponential function.

For example, in the matter of compound interest, the interest rate continues to be the same while the base changes in regular time periods.

Solution

An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we must plug in different values for x and measure the equivalent values for y.

Let's check out the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As demonstrated, the worth of y rise very rapidly as x grows. Imagine 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 goes up from left to right and gets steeper as it persists.

Example 2

Plot the following exponential function:

y = 1/2^x

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

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

If we were to plot 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 smoother as it goes.

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 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 terminology are the powers of an independent variable figure. The common form of an exponential series is:

Source

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