Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

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

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

Here we discuss the fundamentals of an exponential function along with appropriate examples.

What is the formula for an Exponential Function?

The generic formula for an exponential function is f(x) = b^x, where:

1. b is the base, and x is the exponent or power.

2. b is a constant, and x varies

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

How do you graph Exponential Functions?

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

As the exponential function has a constant, it will be necessary to 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 = 1, y will be 2, for x = 2, y will be 4.

According to this method, we get the range values and the domain for the function. Once we have the rate, we need to graph 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 larger than 1, the graph will 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 increasing

• The graph is smooth and constant

• As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis

• As x nears positive infinity, the graph increases without bound.

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

• The graph passes the point (0,1)

• The range is more than 0

• The domain is all real numbers

• The graph is decreasing

• 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 level

• The graph is continuous

Rules

There are a few essential rules to recall when dealing 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 equivalent base, subtract 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 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 note it as (4^x)^3 = 4^(3x).

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

For instance, 1^x = 1 no matter what the rate of x is.

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

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

Examples

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

Example 1

Let’s observe the example of the growing of bacteria. Let’s say we have a group of bacteria that multiples by two hourly, then at the end of hour one, we will have 2 times as many bacteria.

At the end of the second hour, we will have quadruple 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 using 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

Similarly, exponential functions can illustrate exponential decay. If we have a radioactive material that degenerates at a rate of half its volume every hour, then at the end of the first hour, we will have half as much material.

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

At the end of hour three, we will have an eighth as much material (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as follows:

f(t) = 1/2^t

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

As shown, both of these examples use a comparable pattern, which is why 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 depicted by the variable while the base stays the same. Therefore any exponential growth or decline where the base varies is not an exponential function.

For instance, in the case of compound interest, the interest rate stays the same whereas the base is static in normal intervals of time.

Solution

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

Let us look at the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As demonstrated, the worth of y rise very quickly as x rises. Consider we were to plot 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 ,getting steeper as it goes.

Example 2

Plot the following exponential function:

y = 1/2^x

First, let's create a table of values.

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

If we were to draw the x-values and y-values on a coordinate plane, it is going to look like the following:

The above is a decay function. As shown, the graph is a curved line that gets lower 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 unique 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 terms are the powers of an independent variable number. The general form of an exponential series is:

Source

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