Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or rise in a certain base. For instance, let's say a country's population doubles annually. This population growth can be portrayed as an exponential function.
Exponential functions have numerous real-life use cases. Expressed mathematically, an exponential function is written as f(x) = b^x.
In this piece, we will learn the fundamentals of an exponential function along with appropriate 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 a constant, 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 the event where b is higher than 0 and does not equal 1, x will be a real number.
How do you graph Exponential Functions?
To chart an exponential function, we must locate the spots where the function intersects the axes. These are known as the x and y-intercepts.
Since the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To find 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.
According to this method, we achieve the range values and the domain for the function. Once we have 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 properties. When the base of an exponential function is larger than 1, the graph will have the following qualities:
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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 larger 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 smooth and continuous
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As x approaches negative infinity, the graph is asymptomatic towards 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 within 0 and 1, an exponential function presents with the following qualities:
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The graph intersects the point (0,1)
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The range is greater than 0
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The domain is all 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 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 flat
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The graph is constant
Rules
There are several basic rules to recall when engaging 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 example, if we need to divide two exponential functions with 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 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 with a base of 1 is consistently equal to 1.
For instance, 1^x = 1 no matter what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are usually utilized to denote exponential growth. As the variable rises, the value of the function increases quicker and quicker.
Example 1
Let’s observe the example of the growth of bacteria. If we have a culture of bacteria that doubles hourly, then at the end of hour one, we will have 2 times as many bacteria.
At the end of hour two, we will have 4 times as many bacteria (2 x 2).
At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can represent 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 hour one, we will have half as much material.
After two hours, we will have one-fourth as much substance (1/2 x 1/2).
At the end of hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).
This can be represented using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the volume of substance at time t and t is assessed in hours.
As shown, both of these samples follow a similar pattern, which is why they are able to be represented using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base remains constant. Therefore any exponential growth or decline where the base changes is not an exponential function.
For example, in the matter of compound interest, the interest rate continues to be the same whilst the base changes in ordinary time periods.
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 measure the matching values for y.
Let's check out the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the rates of y grow very fast as x grows. Imagine we were to draw 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 and gets steeper as it goes.
Example 2
Plot the following exponential function:
y = 1/2^x
First, let's create a table of values.
As shown, the values of y decrease very quickly as x increases. The reason is because 1/2 is less than 1.
If 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 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 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 expressions are the powers of an independent variable figure. The general form of an exponential series is:
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