June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or increase in a certain base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.

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

Today we will learn the essentials of an exponential function coupled with relevant examples.

What’s the formula for an Exponential Function?

The common 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 is a variable

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 not equal to 1, x will be a real number.

How do you chart Exponential Functions?

To chart an exponential function, we must find the spots where the function intersects the axes. These are called the x and y-intercepts.

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

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

In following this approach, we achieve the range values and the domain for the function. Once we have 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 similar properties. When the base of an exponential function is greater than 1, the graph would have the below qualities:

The line crosses 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 rising
The graph is smooth and continuous
As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
As x advances toward 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 qualities:

The graph passes the point (0,1)
The range is more 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 gets closer to negative infinity, the line approaches without bound
The graph is smooth
The graph is constant

Rules

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

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

For instance, if we need to multiply two exponential functions with a base of 2, then we can note 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 that posses 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 raise 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 forever equivalent 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 despite whatever the value of x is.

Examples

Exponential functions are usually used to indicate exponential growth. As the variable increases, the value of the function grows at a ever-increasing pace.

Example 1

Let’s examine the example of the growing of bacteria. If we have a cluster of bacteria that doubles each hour, then at the close of the first hour, we will have double 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 portrayed using an exponential function as follows:

f(t) = 2^t

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

Example 2

Also, exponential functions can represent exponential decay. If we have a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.

After hour two, we will have one-fourth as much substance (1/2 x 1/2).

At the end of the third hour, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as below:

f(t) = 1/2^t

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

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

In fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base stays fixed. This means that any exponential growth or decomposition where the base changes is not an exponential function.

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

Solution

An exponential function is able to be graphed using 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's check out this 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 worth of y increase very quickly as x grows. Consider we were to draw this exponential function graph on a coordinate plane, it would look like this:

As shown, the graph is a curved line that goes up from left to right and gets steeper as it goes.

Example 2

Graph the following exponential function:

y = 1/2^x

To begin, 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.

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

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

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be written as f(ax)/dx = ax. All derivatives of exponential functions display unique characteristics 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 expressions are the powers of an independent variable number. The common form of an exponential series is:

Source

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