Exponential Functions  Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. For instance, let us suppose a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have many realworld use cases. Expressed mathematically, an exponential function is displayed as f(x) = b^x.
Today we discuss the essentials of an exponential function in conjunction with important examples.
What’s the equation for an Exponential Function?
The general equation for an exponential function is f(x) = b^x, where:

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

b is fixed, and x varies
For example, 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 not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we have to discover the dots where the function crosses the axes. These are referred to as the x and yintercepts.
Since 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 discover the ycoordinates, its essential to set the value for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
According to this method, we achieve the range values and the domain for the function. Once we determine the worth, we need to draw them on the xaxis and the yaxis.
What are the properties of Exponential Functions?
All exponential functions share identical characteristics. When the base of an exponential function is more than 1, the graph will have the following properties:

The line passes 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 smooth and continuous

As x approaches negative infinity, the graph is asymptomatic towards the xaxis

As x advances toward positive infinity, the graph increases without bound.
In events where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following attributes:

The graph passes the point (0,1)

The range is more than 0

The domain is entirely real numbers

The graph is descending

The graph is a curved line

As x advances toward positive infinity, the line in the graph is asymptotic to the xaxis.

As x advances toward negative infinity, the line approaches without bound

The graph is level

The graph is unending
Rules
There are a few vital rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we have 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 an equivalent base, subtract 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^(xy).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For instance, if we have to increase 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 equal 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 equivalent to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are generally used to denote exponential growth. As the variable increases, the value of the function rises quicker and quicker.
Example 1
Let’s observe the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that duplicates each hour, then at the end 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 amount of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive substance that decomposes at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
After the second hour, we will have onefourth as much material (1/2 x 1/2).
At the end of hour three, we will have 1/8 as much substance (1/2 x 1/2 x 1/2).
This can be displayed 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 assessed in hours.
As demonstrated, both of these samples pursue a comparable pattern, which is the reason they can be shown using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base stays the same. This means that any exponential growth or decline where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate stays the same whereas the base is static in ordinary intervals of time.
Solution
An exponential function can be graphed using a table of values. To get the graph of an exponential function, we need to input different values for x and measure the matching values for y.
Let us review this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the rates of y grow very quickly as x grows. Consider we were to draw this exponential function graph on a coordinate plane, it would look like the following:
As shown, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Graph 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 surges. The reason is because 1/2 is less than 1.
If we were to chart the xvalues and yvalues 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 descends 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions exhibit special 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 terms are the powers of an independent variable number. The common form of an exponential series is:
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