Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. For instance, let us suppose a country's population doubles every year. This population growth can be represented as an exponential function.
Exponential functions have numerous real-world use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we discuss the basics of an exponential function in conjunction with relevant examples.
What is the equation 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 a constant, and x is a variable
For instance, 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 the event where b is larger 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 have to 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, we need to set the value for it. Let's focus on the value of b = 2.
To locate the y-coordinates, we need to set the rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this technique, we get the range values and the domain for the function. Once we determine the rate, we need to chart them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable qualities. When the base of an exponential function is more than 1, the graph will have the below 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 level 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 advances toward positive infinity, the graph rises without bound.
In instances 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 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 approaches positive infinity, the line in the 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 a few essential rules to remember 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 with a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an equivalent base, subtract the exponents.
For instance, if we need to divide two exponential functions that have 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 example, if we have to grow 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 with a base of 1 is forever equivalent to 1.
For example, 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 despite whatever the value of x is.
Examples
Exponential functions are generally leveraged to signify exponential growth. As the variable rises, the value of the function grows at a ever-increasing pace.
Example 1
Let’s observe the example of the growing of bacteria. Let’s say we have a culture of bacteria that duplicates hourly, then at the close of the first hour, we will have double as many bacteria.
At the end of the second hour, we will have 4 times 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 displayed utilizing 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
Similarly, exponential functions can portray exponential decay. If we have a radioactive substance that decomposes at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.
At the end of hour two, we will have 1/4 as much substance (1/2 x 1/2).
After three hours, we will have 1/8 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 amount of material at time t and t is calculated in hours.
As demonstrated, both of these illustrations use a comparable pattern, which is why they are able to be depicted using exponential functions.
As a matter of 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 whereas the base remains fixed. Therefore any exponential growth or decomposition where the base varies is not an exponential function.
For instance, in the case of compound interest, the interest rate continues to be the same whereas the base changes in regular intervals of time.
Solution
An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we have to plug in different values for x and asses the corresponding values for y.
Let's review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As shown, the values of y grow very rapidly as x rises. Consider we were to draw this exponential function graph on a coordinate plane, it would look like the following:
As shown, the graph is a curved line that goes up from left to right and gets steeper as it continues.
Example 2
Chart 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 swiftly as x rises. 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 would look like the following:
The above is a decay function. As shown, the graph is a curved line that descends 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 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.
This 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 number. The general form of an exponential series is:
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