Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be portrayed as an exponential function.
Exponential functions have many real-life uses. In mathematical terms, an exponential function is displayed as f(x) = b^x.
In this piece, we discuss the essentials of an exponential function coupled with important examples.
What’s the equation for an Exponential Function?
The generic 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 fixed, and x is a variable
For instance, 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 higher than 0 and not equal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we have to locate the dots where the function crosses the axes. This is 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 rate for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
According to this technique, we get the domain and the range values for the function. After having the worth, 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 qualities. When the base of an exponential function is greater 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 greater 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 advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x approaches positive infinity, the graph grows without bound.
In instances 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 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 advances toward 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 constant
Rules
There are some vital rules to recall when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For instance, if we need 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 the same base, deduct the exponents.
For instance, if we need to divide two exponential functions that have a base of 3, we can compose 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 raise 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 that has a base of 1 is consistently equal to 1.
For example, 1^x = 1 no matter what the rate of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually utilized to indicate exponential growth. As the variable increases, the value of the function rises quicker and quicker.
Example 1
Let’s examine the example of the growing of bacteria. Let us suppose that we have a cluster of bacteria that duplicates hourly, then at the end of the first hour, we will have double as many bacteria.
At the end of hour two, we will have 4x 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 portrayed 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 in hours.
Example 2
Moreover, exponential functions can illustrate exponential decay. Let’s say we had a radioactive substance that decomposes at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.
At the end of hour two, we will have one-fourth as much material (1/2 x 1/2).
At the end of 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 quantity of material at time t and t is measured in hours.
As demonstrated, both of these samples use a comparable pattern, which is why they are able to be represented using exponential functions.
In fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base continues to be the same. 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 stays the same whilst the base changes in ordinary intervals of time.
Solution
An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we must plug in different values for x and then measure the equivalent values for y.
Let us look at this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As demonstrated, the rates of y grow very quickly as x rises. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As you can see, the graph is a curved line that rises from left to right and gets steeper as it goes.
Example 2
Chart 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 surges. This is because 1/2 is less than 1.
Let’s say we were to chart 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 gets lower 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 written as f(ax)/dx = ax. All derivatives of exponential functions present unique features whereby 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 digit. The general form of an exponential series is:
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