Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or rise in a particular 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-life applications. Expressed mathematically, an exponential function is written as f(x) = b^x.
Here we will review the fundamentals of an exponential function coupled with important examples.
What’s 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 is a variable
As an illustration, 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 cases where b is greater than 0 and not equal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we must discover the points where the function crosses the axes. This is called the x and y-intercepts.
As the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, its essential 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 approach, we get the range values and the domain for the function. After having the values, 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 characteristics:
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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 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 ongoing
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As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
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As x nears 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 exhibits the following characteristics:
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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 descending
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The graph is a curved line
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As x approaches positive infinity, the line within graph is asymptotic to the x-axis.
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As x gets closer to negative infinity, the line approaches without bound
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The graph is flat
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The graph is continuous
Rules
There are some basic rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we have to multiply two exponential functions with 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 have to divide two exponential functions with a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For example, if we have to raise 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 consistently equal 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 generally used to signify 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 group of bacteria that doubles every hour, then at the end of the first hour, 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 8 times 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 hourly.
Example 2
Also, exponential functions can portray exponential decay. If we have a radioactive material that degenerates at a rate of half its volume 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).
After hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).
This can be displayed 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 examples use a similar pattern, which is why they can be depicted using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains the same. This indicates that any exponential growth or decomposition 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 varies in ordinary intervals of time.
Solution
An exponential function is able to be graphed utilizing 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's review 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 fast as x rises. 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 continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To start, let's draw up a table of values.
As you can see, the values of y decrease very quickly as x increases. The reason is because 1/2 is less than 1.
Let’s say we were to draw the x-values and y-values on a coordinate plane, it is going to look like this:
This is a decay function. As shown, the graph is a curved line that gets lower from right to left and gets flatter as it proceeds.
The Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x can be shown 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 terminology are the powers of an independent variable number. The general form of an exponential series is:
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