Exponential Functions  Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or increase in a particular base. Take this, for example, let's say a country's population doubles every year. This population growth can be depicted in the form of an exponential function.
Exponential functions have many reallife uses. Expressed mathematically, an exponential function is shown as f(x) = b^x.
Here we discuss the fundamentals of an exponential function coupled with important 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
As an illustration, 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 cases where b is higher than 0 and unequal to 1, x will be a real number.
How do you graph Exponential Functions?
To plot an exponential function, we must locate the dots where the function intersects the axes. These are 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 instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this approach, we determine the range values and the domain for the function. Once we have the worth, we need to draw them on the xaxis and the yaxis.
What are the properties of Exponential Functions?
All exponential functions share similar characteristics. When the base of an exponential function is more than 1, the graph will have the below characteristics:

The line passes the point (0,1)

The domain is all positive real numbers

The range is larger than 0

The graph is a curved line

The graph is rising

The graph is smooth and ongoing

As x advances toward negative infinity, the graph is asymptomatic regarding the xaxis

As x nears positive infinity, the graph increases without bound.
In situations where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following characteristics:

The graph passes the point (0,1)

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

As x approaches negative infinity, the line approaches without bound

The graph is smooth

The graph is unending
Rules
There are some basic rules to recall 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 with 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, deduct the exponents.
For instance, if we need to divide two exponential functions that posses a base of 3, we can write it as 3^x / 3^y = 3^(xy).
Rule 3: To increase 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, we are able to write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is consistently equivalent to 1.
For instance, 1^x = 1 no matter 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 commonly used to indicate exponential growth. As the variable grows, the value of the function grows quicker and quicker.
Example 1
Let’s examine the example of the growing of bacteria. If we have a cluster of bacteria that duplicates hourly, then at the end of hour one, we will have double as many bacteria.
At the end of hour two, we will have 4 times as many bacteria (2 x 2).
At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be represented 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 in hours.
Example 2
Moreover, exponential functions can portray exponential decay. Let’s say we had a radioactive substance that decomposes at a rate of half its quantity every hour, then at the end of hour one, we will have half as much substance.
At the end of hour two, we will have a quarter 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 represented using an exponential equation as below:
f(t) = 1/2^t
where f(t) is the quantity of material at time t and t is calculated in hours.
As demonstrated, both of these illustrations use a comparable pattern, which is the reason they can be depicted using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base remains constant. Therefore any exponential growth or decay where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate continues to be the same while the base is static in normal amounts of time.
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 asses the corresponding values for y.
Let's review this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the values of y increase very quickly as x grows. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it persists.
Example 2
Draw 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 quickly as x surges. The reason is because 1/2 is less than 1.
If we were to graph the xvalues and yvalues on a coordinate plane, it is going to look like this:
The above is a decay function. As shown, the graph is a curved line that decreases from right to left and gets flatter as it continues.
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 exhibit unique features 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:
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