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's say a country's population doubles every year. This population growth can be portrayed as an exponential function.
Exponential functions have multiple reallife uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we discuss the essentials of an exponential function along with appropriate examples.
What’s the equation 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 fixed, and x is a variable
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 a situation where b is greater than 0 and does not equal 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we need to locate the dots where the function crosses the axes. This is called the x and yintercepts.
Considering the fact that 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 locate the ycoordinates, one must to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
According to this method, we achieve the domain and the range values for the function. Once we have the worth, we need to graph them on the xaxis and the yaxis.
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 is going to 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 rising

The graph is smooth and constant

As x nears negative infinity, the graph is asymptomatic regarding the xaxis

As x advances toward 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 characteristics:

The graph crosses the point (0,1)

The range is larger 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 within graph is asymptotic to the xaxis.

As x approaches negative infinity, the line approaches without bound

The graph is flat

The graph is continuous
Rules
There are several vital rules to recall when working with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For instance, if we have to multiply two exponential functions that posses a base of 2, then we can write 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 need to divide two exponential functions that have 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 example, if we have to raise 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 with a base of 1 is always 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 identical to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are commonly utilized to indicate exponential growth. As the variable grows, the value of the function increases faster and faster.
Example 1
Let's look at the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that duplicates every hour, then at the close of hour one, we will have 2 times as many bacteria.
At the end of hour two, we will have 4x 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 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 hourly.
Example 2
Also, exponential functions can illustrate exponential decay. If we have a radioactive substance that decays 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 a quarter as much material (1/2 x 1/2).
At the end of three hours, 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 quantity of substance at time t and t is assessed in hours.
As demonstrated, both of these samples use a comparable pattern, which is why they are able to be represented using exponential functions.
In fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base stays the same. This indicates that any exponential growth or decomposition where the base changes is not an exponential function.
For instance, in the case of compound interest, the interest rate continues to be the same whereas the base is static in ordinary amounts of time.
Solution
An exponential function can be graphed utilizing 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 check out 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 grow very rapidly as x increases. If we were to plot 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 goes up from left to right ,getting steeper as it persists.
Example 2
Plot the following exponential function:
y = 1/2^x
To begin, let's draw up a table of values.
As shown, the values of y decrease very quickly as x rises. This is because 1/2 is less than 1.
If we were to chart the xvalues and yvalues on a coordinate plane, it is going to 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 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 exhibit particular features where 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 expressions are the powers of an independent variable digit. The common form of an exponential series is:
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