Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or rise in a particular base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be depicted as an exponential function.
Exponential functions have numerous real-world applications. In mathematical terms, an exponential function is shown as f(x) = b^x.
In this piece, we discuss the essentials 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:
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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 instance, 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 chart Exponential Functions?
To chart an exponential function, we must find the points where the function intersects the axes. These are known as the x and y-intercepts.
Considering the fact that the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To locate the y-coordinates, its essential to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this technique, we determine the range values and the domain for the function. Once we determine the values, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share identical qualities. When the base of an exponential function is larger than 1, the graph will have the below properties:
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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 more 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 level and constant
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As x nears 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 situations where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following characteristics:
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The graph passes the point (0,1)
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The range is more 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 advances toward 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 level
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The graph is unending
Rules
There are some vital rules to recall when working 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 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, subtract the exponents.
For instance, if we have to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
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, then we can write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is always 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 despite whatever the value of x is.
Examples
Exponential functions are usually used to signify exponential growth. As the variable increases, the value of the function increases at a ever-increasing pace.
Example 1
Let’s examine the example of the growing of bacteria. Let’s say we have a cluster of bacteria that doubles hourly, then at the end 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 the third hour, we will have 8x 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 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 amount every hour, then at the end of one hour, we will have half as much material.
At the end of the second hour, we will have 1/4 as much substance (1/2 x 1/2).
At the end of hour three, we will have an eighth as much material (1/2 x 1/2 x 1/2).
This can be shown 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 calculated in hours.
As demonstrated, both of these samples follow 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. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base continues to be the same. This means that any exponential growth or decomposition where the base changes is not an exponential function.
For instance, in the scenario of compound interest, the interest rate continues to be the same whilst the base changes in ordinary time periods.
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 then calculate the corresponding 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 rates of y increase very quickly as x rises. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As shown, the graph is a curved line that rises from left to right and gets steeper as it persists.
Example 2
Graph 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 swiftly as x rises. The reason 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 descends from right to left and gets flatter as it proceeds.
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 special properties 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 expressions are the powers of an independent variable digit. The common form of an exponential series is:
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