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 particular base. Take this, for example, let us assume a country's population doubles every year. This population growth can be portrayed as an exponential function.

Exponential functions have multiple real-world uses. Expressed mathematically, an exponential function is shown as f(x) = b^x.

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

What’s the formula 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 varies

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

How do you plot Exponential Functions?

To chart an exponential function, we must find the dots where the function intersects the axes. This is called 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 locate the y-coordinates, one must to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.

According to this technique, we achieve the domain and the range values for the function. Once we determine the worth, we need to draw them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical properties. When the base of an exponential function is larger than 1, the graph will have the below properties:

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 on an incline
The graph is flat and ongoing
As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
As x nears positive infinity, the graph rises without bound.

In situations where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following properties:

The graph crosses the point (0,1)
The range is greater than 0
The domain is all 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 smooth
The graph is constant

Rules

There are several basic rules to recall 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 that posses a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with an identical base, deduct the exponents.

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

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

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

For instance, 1^x = 1 regardless of what the worth of x is.

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

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

Examples

Exponential functions are usually leveraged to indicate exponential growth. As the variable rises, the value of the function grows faster and faster.

Example 1

Let’s observe the example of the growth of bacteria. Let’s say we have a culture of bacteria that multiples by two every hour, 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 displayed 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

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

At the end of the second hour, we will have a quarter as much substance (1/2 x 1/2).

After hour three, we will have an eighth as much material (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 material at time t and t is calculated in hours.

As you can see, both of these examples pursue 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. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base continues to be constant. This means that any exponential growth or decay where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate remains the same whilst the base is static in ordinary time periods.

Solution

An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we need to input different values for x and then measure the matching values for y.

Let us check out 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 rates of y increase very quickly as x grows. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like the following:

As you can see, the graph is a curved line that rises from left to right ,getting steeper as it continues.

Example 2

Chart 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. This 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 what you see below:

This is a decay function. As you can see, the graph is a curved line that decreases 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions present particular properties whereby 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 digit. The general form of an exponential series is:

Source

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