June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

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

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

In this piece, we will review the basics of an exponential function in conjunction with important examples.

What’s 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 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 cases where b is greater than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we have to discover the spots where the function crosses the axes. This is known as the x and y-intercepts.

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

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

In following this approach, we determine the domain and the range values for the function. After having the values, 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 properties. When the base of an exponential function is more 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 rising
The graph is level and continuous
As x advances toward negative infinity, the graph is asymptomatic towards the x-axis
As x nears positive infinity, the graph rises without bound.

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

The graph intersects the point (0,1)
The range is more than 0
The domain is entirely real numbers
The graph is declining
The graph is a curved line
As x approaches 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 smooth
The graph is constant

Rules

There are several basic rules to recall when working with exponential functions.

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

For example, if we need 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 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 increase an exponential function to a power, multiply the exponents.

For instance, if we have to grow 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 with a base of 1 is forever equivalent to 1.

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

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

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

Examples

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

Example 1

Let's look at the example of the growing of bacteria. Let’s say we have a group of bacteria that doubles each hour, then at the close of the first hour, we will have twice as many bacteria.

At the end of hour two, we will have quadruple 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 portrayed 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

Moreover, exponential functions can represent exponential decay. Let’s say we had a dangerous substance that degenerates at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.

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

After the third hour, we will have one-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 substance at time t and t is measured in hours.

As demonstrated, both of these examples use a comparable pattern, which is the reason they are able to be represented 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 represented by the variable whereas the base remains constant. This means that any exponential growth or decay where the base changes is not an exponential function.

For instance, in the scenario of compound interest, the interest rate stays the same while the base is static in ordinary amounts of time.

Solution

An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we must input different values for x and asses the matching values for y.

Let us review this example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As demonstrated, the rates of y rise very fast as x grows. Imagine 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 rises from left to right and gets steeper as it persists.

Example 2

Plot the following exponential function:

y = 1/2^x

To start, let's create a table of values.

As shown, the values of y decrease very rapidly as x rises. This is because 1/2 is less than 1.

Let’s say we were to plot the x-values and y-values on a coordinate plane, it is going to look like the following:

This is a decay function. As shown, 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions display special features whereby 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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