Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a certain base. For instance, let us assume a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have numerous real-world uses. Expressed mathematically, an exponential function is written as f(x) = b^x.
In this piece, we will review the basics of an exponential function in conjunction with appropriate examples.
What is the formula for an Exponential Function?
The generic equation 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 fixed, 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 not equal to 1, x will be a real number.
How do you graph Exponential Functions?
To graph an exponential function, we must find the dots where the function intersects the axes. These are called the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, its essential to set the rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
According to this approach, we determine the domain and the range values for the function. Once we have the worth, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar characteristics. When the base of an exponential function is greater than 1, the graph is going to have the below qualities:
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The line crosses 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 flat and continuous
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As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x advances toward positive infinity, the graph increases without bound.
In cases where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following qualities:
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The graph intersects the point (0,1)
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The range is more than 0
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The domain is entirely real numbers
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The graph is descending
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The graph is a curved line
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As x advances toward positive infinity, the line within 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 flat
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The graph is unending
Rules
There are a few basic rules to recall when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For example, 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, subtract the exponents.
For example, if we have 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 grow 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, we are able to note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equivalent to 1.
For example, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are generally leveraged to signify exponential growth. As the variable grows, the value of the function rises faster and faster.
Example 1
Let's look at the example of the growing of bacteria. If we have a cluster of bacteria that multiples by two hourly, then at the close of the first hour, we will have twice 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 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 number of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can illustrate exponential decay. If we have a radioactive substance that degenerates 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 a quarter as much substance (1/2 x 1/2).
After three hours, we will have 1/8 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 volume of material at time t and t is assessed in hours.
As shown, both of these illustrations use a comparable pattern, which is the reason they are able to be shown using exponential functions.
In fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains constant. 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 remains 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 must input different values for x and measure the matching values for y.
Let's check out the following example.
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 increase very rapidly as x increases. Consider we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that goes up from left to right and gets steeper as it persists.
Example 2
Plot the following exponential function:
y = 1/2^x
To start, let's draw up 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 draw the x-values and y-values on a coordinate plane, it is going to look like what you see below:
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 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 present unique features whereby 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 digit. The general form of an exponential series is:
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