Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a certain base. For example, let us assume a country's population doubles yearly. This population growth can be represented in the form of an exponential function.
Exponential functions have numerous real-world applications. Expressed mathematically, an exponential function is displayed as f(x) = b^x.
Today we will review the essentials of an exponential function along with relevant examples.
What’s the formula for an Exponential Function?
The general 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 fixed, and x varies
As an illustration, 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 cases where b is larger than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we must find the spots where the function intersects the axes. These are called the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To discover the y-coordinates, we need to set the value for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
According to this approach, we determine the range values and the domain for the function. After having the worth, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar characteristics. When the base of an exponential function is larger than 1, the graph will have the following 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 on an incline
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The graph is flat and constant
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As x approaches negative infinity, the graph is asymptomatic regarding the x-axis
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As x nears positive infinity, the graph rises without bound.
In cases where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:
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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 declining
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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 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 a few essential rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we have to multiply two exponential functions that have 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 identical base, deduct 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 example, if we have to increase an exponential function with a base of 4 to the third power, we are able to write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equivalent to 1.
For instance, 1^x = 1 regardless of 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 no matter what the value of x is.
Examples
Exponential functions are usually utilized to indicate exponential growth. As the variable rises, 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 group of bacteria that duplicates hourly, then at the close 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 the third hour, 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 total sum of bacteria at time t and t is measured hourly.
Example 2
Similarly, exponential functions can portray exponential decay. If we have a dangerous substance 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.
After the second hour, we will have 1/4 as much material (1/2 x 1/2).
At the end of the third hour, we will have an eighth 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 quantity of substance at time t and t is calculated in hours.
As shown, both of these samples use a comparable pattern, which is why they can be shown using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable while the base stays constant. This means that any exponential growth or decline where the base changes is not an exponential function.
For instance, in the case of compound interest, the interest rate continues to be the same whilst the base is static in ordinary amounts of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to plug in different values for x and then measure the matching values for y.
Let us look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the worth of y increase very quickly as x grows. Imagine 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 rises from left to right ,getting steeper as it persists.
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.
Let’s say we were to graph the x-values and y-values on a coordinate plane, it would look like this:
The above is a decay function. As you can see, 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 exhibit special characteristics 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:
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