Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a particular base. Take this, for example, let's say a country's population doubles annually. This population growth can be represented in the form of an exponential function.
Exponential functions have multiple real-life uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we discuss the basics of an exponential function along with relevant examples.
What’s the equation for an Exponential Function?
The generic 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 is a variable
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 not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we have to find the dots where the function crosses the axes. These are referred to as the x and y-intercepts.
As 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 locate the y-coordinates, one must to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this method, we determine the range values and the domain for the function. Once we determine the values, we need to draw 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 will 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 on an incline
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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 advances toward positive infinity, the graph grows without bound.
In events where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following characteristics:
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The graph intersects 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 advances toward 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 unending
Rules
There are some vital 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 need to multiply two exponential functions with 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 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 grow an exponential function to a power, multiply the exponents.
For example, if we have to grow 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 always equivalent to 1.
For instance, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are usually utilized to signify exponential growth. As the variable rises, the value of the function increases faster and faster.
Example 1
Let’s examine the example of the growing of bacteria. If we have a cluster of bacteria that doubles each hour, then at the close of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x 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 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 in hours.
Example 2
Moreover, exponential functions can illustrate exponential decay. If we have a dangerous substance that degenerates at a rate of half its amount 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 one-fourth as much material (1/2 x 1/2).
At the end of the third hour, 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 assessed 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. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base remains constant. This indicates that any exponential growth or decay where the base changes is not an exponential function.
For example, in the scenario of compound interest, the interest rate continues to be the same while the base varies in ordinary time periods.
Solution
An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we need to enter different values for x and then calculate the corresponding values for y.
Let's look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the rates of y grow very rapidly as x increases. If 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 ,getting steeper as it goes.
Example 2
Chart the following exponential function:
y = 1/2^x
To begin, let's make 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 smoother as it continues.
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 particular characteristics by which 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 figure. The general form of an exponential series is:
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