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's say a country's population doubles annually. This population growth can be depicted as an exponential function.
Exponential functions have numerous real-world use cases. Expressed mathematically, an exponential function is shown as f(x) = b^x.
In this piece, we will review the basics of an exponential function coupled with important examples.
What is the formula for an Exponential Function?
The common 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 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 a situation where b is larger than 0 and unequal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we must find the points where the function intersects the axes. This is known as 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 locate the y-coordinates, we need 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 approach, we get the domain and the range values for the function. Once we have the worth, we need to graph them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable characteristics. When the base of an exponential function is larger than 1, the graph would have the following qualities:
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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 larger than 0
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The graph is a curved line
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The graph is increasing
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The graph is smooth and continuous
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As x advances toward negative infinity, the graph is asymptomatic regarding 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 between 0 and 1, an exponential function exhibits the following properties:
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The graph intersects 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 nears positive infinity, the line in the graph is asymptotic to the x-axis.
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As x approaches negative infinity, the line approaches without bound
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The graph is smooth
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The graph is constant
Rules
There are a few basic rules to recall when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we need to multiply two exponential functions that have 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 example, if we need to divide two exponential functions with a base of 3, we can compose 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 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 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 equivalent to 0.
For instance, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are generally utilized to indicate exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.
Example 1
Let’s examine the example of the growth of bacteria. Let’s say we have a culture of bacteria that doubles every hour, then at the close 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 the third hour, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed using an exponential function as follows:
f(t) = 2^t
where f(t) is the total sum of bacteria at time t and t is measured hourly.
Example 2
Similarly, exponential functions can portray exponential decay. Let’s say we had a dangerous material that decays 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 one-fourth as much material (1/2 x 1/2).
After three hours, we will have one-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 amount of material at time t and t is measured in hours.
As shown, both of these examples use a comparable pattern, which is the reason they are able to be shown using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is represented by the variable while the base stays fixed. This means that any exponential growth or decay where the base varies is not an exponential function.
For instance, in the case of compound interest, the interest rate remains the same while the base is static in regular time periods.
Solution
An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we have to plug in different values for x and measure the corresponding values for y.
Let's look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the worth of y rise very quickly as x grows. If we were to draw 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 rises from left to right ,getting steeper as it continues.
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 rises. The reason is because 1/2 is less than 1.
If we were to graph the x-values and y-values on a coordinate plane, it is going to look like what you see below:
This is a decay function. As shown, the graph is a curved line that decreases from right to left and gets flatter 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 particular properties 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 figure. The common form of an exponential series is:
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