Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or rise in a certain base. Take this, for example, let us assume a country's population doubles every year. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-world use cases. In mathematical terms, an exponential function is displayed as f(x) = b^x.
In this piece, we discuss the fundamentals of an exponential function in conjunction with important examples.
What’s the formula for an Exponential Function?
The general formula for an exponential function is f(x) = b^x, where:
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b is the base, and x is the exponent or power.
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b is a constant, 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 cases where b is higher than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we need to locate the spots where the function crosses the axes. This is called the x and y-intercepts.
Since the exponential function has a constant, we need to set the value for it. Let's focus on the value of b = 2.
To discover the y-coordinates, one must to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
According to this method, we determine the range values and the domain for the function. Once we have the worth, we need to draw 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 larger than 1, the graph would have the following characteristics:
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The line intersects the point (0,1)
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The domain is all positive real numbers
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The range is greater than 0
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The graph is a curved line
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The graph is rising
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The graph is level and ongoing
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As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x approaches positive infinity, the graph rises without bound.
In events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following characteristics:
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The graph passes the point (0,1)
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The range is greater than 0
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The domain is all real numbers
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The graph is decreasing
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The graph is a curved line
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As x approaches positive infinity, the line in the graph is asymptotic to the x-axis.
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As x gets closer to negative infinity, the line approaches without bound
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The graph is smooth
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The graph is unending
Rules
There are some basic rules to remember when dealing 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 with 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, deduct the exponents.
For instance, if we need to divide two exponential functions with a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For instance, 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 forever equivalent to 1.
For example, 1^x = 1 no matter 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 utilized to denote exponential growth. As the variable rises, the value of the function rises at a ever-increasing pace.
Example 1
Let's look at the example of the growing of bacteria. If we have a culture of bacteria that duplicates every hour, then at the close of hour one, 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 8x 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 in hours.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a dangerous material that degenerates at a rate of half its amount every hour, then at the end of the first hour, we will have half as much material.
After hour two, we will have a quarter as much material (1/2 x 1/2).
After hour three, we will have an eighth 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 assessed in hours.
As you can see, both of these samples follow a similar pattern, which is why they are able to be shown using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains fixed. Therefore any exponential growth or decline where the base varies is not an exponential function.
For instance, in the matter of compound interest, the interest rate remains the same whereas the base changes in ordinary amounts of time.
Solution
An exponential function can be graphed using a table of values. To get the graph of an exponential function, we have to enter different values for x and calculate the equivalent values for y.
Let us look at the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the values of y increase very quickly as x grows. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like this:
As seen above, the graph is a curved line that rises from left to right and gets steeper as it goes.
Example 2
Plot the following exponential function:
y = 1/2^x
To start, let's create a table of values.
As you can see, the values of y decrease very quickly as x increases. This is because 1/2 is less than 1.
If we were to chart 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 you can see, the graph is a curved line that decreases from right to left and gets smoother as it continues.
The Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x can be written as f(ax)/dx = ax. All derivatives of exponential functions exhibit special features by which 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 common form of an exponential series is:
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