Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a specific base. For example, let us suppose a country's population doubles annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have many real-world uses. In mathematical terms, an exponential function is displayed as f(x) = b^x.
In this piece, we will learn the basics of an exponential function coupled 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 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 the event where b is larger than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we need to locate the dots 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, it will be necessary to set the value for it. Let's focus on the value of b = 2.
To discover the y-coordinates, its essential to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
By following this method, we achieve 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 identical characteristics. When the base of an exponential function is more than 1, the graph is going to have the following qualities:
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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 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 smooth and constant
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As x approaches 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 events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following attributes:
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The graph passes the point (0,1)
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The range is greater 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 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 flat
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The graph is unending
Rules
There are several basic rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we have 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 the same 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 grow 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, 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 instance, 1^x = 1 regardless of what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly used to signify exponential growth. As the variable increases, the value of the function grows quicker and quicker.
Example 1
Let’s observe the example of the growth of bacteria. Let’s say we have a group of bacteria that duplicates hourly, then at the end of hour one, we will have twice 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 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 amount of bacteria at time t and t is measured in hours.
Example 2
Also, exponential functions can represent exponential decay. Let’s say we had a dangerous substance that decomposes 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 two hours, we will have one-fourth as much substance (1/2 x 1/2).
After the third hour, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).
This can be represented using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the quantity of material at time t and t is assessed in hours.
As shown, both of these samples use a comparable pattern, which is the reason they are able to be depicted using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable while the base remains the same. Therefore any exponential growth or decay where the base is different is not an exponential function.
For example, in the scenario of compound interest, the interest rate continues to be the same while the base changes in regular time periods.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we need to plug in different values for x and measure the equivalent values for y.
Let us look at the example below.
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 grow very quickly as x rises. 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 ,getting steeper as it goes.
Example 2
Plot the following exponential function:
y = 1/2^x
First, let's draw up a table of values.
As shown, the values of y decrease very swiftly as x surges. This is because 1/2 is less than 1.
If we were to graph the x-values and y-values on a coordinate plane, it is going to look like what you see below:
This is a decay function. As shown, the graph is a curved line that decreases 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions exhibit particular characteristics where 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 terminology are the powers of an independent variable number. The general form of an exponential series is:
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