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 instance, let us suppose a country's population doubles annually. This population growth can be represented in the form of an exponential function.
Exponential functions have multiple real-world use cases. Expressed mathematically, an exponential function is shown as f(x) = b^x.
In this piece, we discuss the basics of an exponential function in conjunction with relevant examples.
What is the formula for an Exponential Function?
The general 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 is a variable
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 the event where b is larger 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 need to find the dots where the function crosses the axes. This is referred to 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 focus on the value of b = 2.
To locate the y-coordinates, we need to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
In following this approach, we determine the domain and the range values for the function. Once we determine the values, we need to chart 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 more than 1, the graph would have the following characteristics:
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The line passes 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 smooth and continuous
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As x approaches negative infinity, the graph is asymptomatic concerning 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 between 0 and 1, an exponential function displays the following properties:
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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 declining
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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 approaches negative infinity, the line approaches without bound
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The graph is smooth
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The graph is continuous
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 instance, if we have to multiply two exponential functions that posses a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, subtract the exponents.
For instance, if we have to divide two exponential functions with a base of 3, we can write it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For instance, if we have to grow 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 consistently equivalent to 1.
For instance, 1^x = 1 no matter what the rate of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually utilized to signify exponential growth. As the variable increases, the value of the function grows faster and faster.
Example 1
Let’s examine the example of the growth of bacteria. Let’s say we have a cluster of bacteria that multiples by two 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 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be displayed 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
Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of one hour, we will have half as much substance.
At the end of 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 represented 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 shown, both of these illustrations follow a similar pattern, which is the reason they can be depicted using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base stays constant. Therefore any exponential growth or decay where the base is different is not an exponential function.
For instance, in the scenario of compound interest, the interest rate continues to be the same while the base is static 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 must input different values for x and then asses the equivalent 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 demonstrated, the values of y rise very quickly as x rises. Imagine we were to plot 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 and gets steeper as it continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To begin, let's draw up a table of values.
As you can see, the values of y decrease very rapidly as x rises. This is because 1/2 is less than 1.
Let’s say we were to plot the x-values and y-values on a coordinate plane, it would look like what you see below:
The above is a decay function. As you can see, 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 present special features 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 terminology are the powers of an independent variable digit. The common form of an exponential series is:
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