Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a certain base. For example, let's say a country's population doubles annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have many real-life uses. Expressed mathematically, an exponential function is shown as f(x) = b^x.
Here we discuss the basics of an exponential function along with relevant examples.
What is the equation for an Exponential Function?
The generic 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
For example, 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 higher than 0 and not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we must discover the dots where the function crosses the axes. These are called the x and y-intercepts.
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 y-coordinates, one must to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
According to this method, we get 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 similar qualities. When the base of an exponential function is greater than 1, the graph is going to have the following characteristics:
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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 larger than 0
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The graph is a curved line
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The graph is on an incline
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The graph is smooth and constant
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As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x nears positive infinity, the graph grows 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 attributes:
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The graph crosses the point (0,1)
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The range is larger than 0
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The domain is all 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 nears positive infinity, the line within graph is asymptotic to the x-axis.
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As x advances toward negative infinity, the line approaches without bound
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The graph is flat
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The graph is constant
Rules
There are several basic rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we need to multiply two exponential functions that have 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 that have 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 raise an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equal 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 example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are generally leveraged to signify exponential growth. As the variable rises, the value of the function increases faster and faster.
Example 1
Let’s examine the example of the growth of bacteria. Let us suppose that we have a culture of bacteria that multiples by two hourly, then at the end 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 hour three, we will have 8 times 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 in hours.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive material that decomposes at a rate of half its amount every hour, then at the end of the first hour, we will have half as much substance.
At the end of the second hour, we will have one-fourth as much material (1/2 x 1/2).
After hour three, we will have one-eighth as much material (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 measured in hours.
As demonstrated, both of these examples pursue a comparable pattern, which is why they are able to be represented using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base continues to be fixed. This indicates that any exponential growth or decomposition where the base is different is not an exponential function.
For example, in the matter of compound interest, the interest rate remains the same whereas the base is static in ordinary amounts of time.
Solution
An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we need to plug in different values for x and asses the matching values for y.
Let's review this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As you can see, the rates of y increase very quickly as x increases. Consider we were to plot this exponential function graph on a coordinate plane, it would look like this:
As you can see, the graph is a curved line that rises from left to right and gets steeper as it goes.
Example 2
Draw the following exponential function:
y = 1/2^x
To start, let's create a table of values.
As shown, 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 this:
The above is a decay function. As shown, the graph is a curved line that descends from right to left and gets smoother as it proceeds.
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 particular properties 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 number. The general form of an exponential series is:
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