Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a specific base. For example, let us assume a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have numerous real-life use cases. Expressed mathematically, an exponential function is written as f(x) = b^x.
Today we will learn the basics of an exponential function along with appropriate examples.
What is the formula 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 varies
For example, 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 a situation where b is larger than 0 and does not equal 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we must find the spots where the function intersects the axes. These are referred to as 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, one must to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
In following this method, we get the range values and the domain for the function. Once we determine 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 comparable properties. When the base of an exponential function is larger than 1, the graph is going to have the below characteristics:
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The line crosses 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 smooth and constant
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As x nears negative infinity, the graph is asymptomatic towards the x-axis
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As x approaches positive infinity, the graph rises without bound.
In events where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:
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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 approaches positive infinity, the line in the 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 smooth
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The graph is continuous
Rules
There are several basic rules to bear in mind 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 an identical 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 grow an exponential function with a base of 4 to the third power, we are able to compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is forever equal 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 equivalent to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually used to denote 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. If we have a group of bacteria that duplicates every hour, then at the close of hour one, we will have double as many bacteria.
At the end of hour two, 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 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 represent exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its amount every hour, then at the end of hour one, we will have half as much substance.
After the second hour, we will have 1/4 as much substance (1/2 x 1/2).
At the end of the third hour, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).
This can be shown using an exponential equation as below:
f(t) = 1/2^t
where f(t) is the amount of substance at time t and t is calculated in hours.
As shown, both of these examples use a comparable pattern, which is the reason they are able to be represented 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 while the base continues to be the same. This indicates that any exponential growth or decomposition where the base changes is not an exponential function.
For example, in the case of compound interest, the interest rate stays the same whereas the base is static in ordinary amounts 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 need to plug in different values for x and calculate the matching values for y.
Let us check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As you can see, the values of y rise very fast as x increases. Imagine we were to plot 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
Chart the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As you can see, the values of y decrease very swiftly as x increases. The reason 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:
This is a decay function. As you can see, the graph is a curved line that descends from right to left and gets flatter 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 display particular properties 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 figure. The common form of an exponential series is:
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