Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. For instance, 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. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
In this piece, we discuss the essentials of an exponential function along with appropriate examples.
What is the equation 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 fixed, and x varies
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 unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we must locate the points where the function crosses the axes. These are called the x and y-intercepts.
As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.
To discover the y-coordinates, one must 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 get the domain and the range values for the function. After having 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 characteristics. When the base of an exponential function is more than 1, the graph would have the following 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 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 flat 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 increases without bound.
In situations where the bases are fractions or decimals between 0 and 1, an exponential function displays the following characteristics:
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The graph passes the point (0,1)
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The range is larger than 0
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The domain is entirely real numbers
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The graph is decreasing
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The graph is a curved line
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As x nears 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 level
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The graph is constant
Rules
There are a few basic rules to bear in mind 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 write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an equivalent base, subtract the exponents.
For example, if we need to divide two exponential functions that have a base of 3, we can write 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 increase an exponential function with a base of 4 to the third power, then we can compose 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 regardless of what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For example, 0^x = 0 no matter what 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 faster and faster.
Example 1
Let's look at the example of the growing of bacteria. Let’s say we have a culture of bacteria that duplicates each hour, then at the close of the first hour, we will have double as many bacteria.
At the end of hour two, we will have quadruple as many bacteria (2 x 2).
At the end of hour three, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be represented 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 illustrate exponential decay. Let’s say we had a radioactive material that degenerates at a rate of half its amount every hour, then at the end of hour one, we will have half as much substance.
After two hours, 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 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 substance at time t and t is calculated in hours.
As shown, both of these examples use a similar pattern, which is why they can be shown using exponential functions.
In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base remains the same. This indicates that any exponential growth or decay where the base changes is not an exponential function.
For instance, in the case of compound interest, the interest rate stays the same whilst the base varies in ordinary intervals of time.
Solution
An exponential function can be graphed using a table of values. To get the graph of an exponential function, we must enter different values for x and then calculate the corresponding values for y.
Let us check out the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As you can see, the rates of y rise very fast as x increases. 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 rises from left to right ,getting steeper as it continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As shown, the values of y decrease very rapidly as x surges. The reason 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 what you see below:
The above is a decay function. As shown, the graph is a curved line that descends from right to left and gets smoother as it goes.
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 present particular characteristics 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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