Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. For instance, let us suppose a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have numerous real-world use cases. In mathematical terms, an exponential function is displayed as f(x) = b^x.
Today we will learn the fundamentals of an exponential function in conjunction with appropriate examples.
What is the formula for an Exponential Function?
The general 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 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 cases where b is higher than 0 and does not equal 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we have to discover the spots where the function crosses the axes. This is known as the x and y-intercepts.
As the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To discover the y-coordinates, we need to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
According to this method, we get the range values and the domain for the function. After having the worth, we need to graph them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable characteristics. When the base of an exponential function is larger than 1, the graph would 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 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 smooth and ongoing
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As x nears negative infinity, the graph is asymptomatic concerning the x-axis
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As x nears positive infinity, the graph increases without bound.
In events where the bases are fractions or decimals between 0 and 1, an exponential function displays the following characteristics:
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The graph passes 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 declining
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The graph is a curved line
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As x nears 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 level
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The graph is unending
Rules
There are several basic rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For example, if we have to multiply two exponential functions that have a base of 2, then we can note 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 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 example, if we have to increase an exponential function with a base of 4 to the third power, then we can note 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 regardless of what the rate of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are commonly utilized to denote exponential growth. As the variable grows, the value of the function increases quicker and quicker.
Example 1
Let's look at the example of the growth of bacteria. Let’s say we have a group of bacteria that doubles each hour, then at the close of the first hour, 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 hour three, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed 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 illustrate exponential decay. Let’s say we had a radioactive material that decomposes at a rate of half its quantity every hour, then at the end of hour one, we will have half as much substance.
At the end of hour two, we will have 1/4 as much substance (1/2 x 1/2).
After the third hour, we will have an eighth 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 volume of material at time t and t is calculated in hours.
As demonstrated, both of these illustrations follow a comparable pattern, which is why they can be represented using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable while the base stays the same. This means 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 stays the same whilst the base varies in ordinary time periods.
Solution
An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we must input different values for x and measure the corresponding 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 rates of y rise very rapidly as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Draw 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 swiftly 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 this:
The above 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 shown as f(ax)/dx = ax. All derivatives of exponential functions present special properties by which 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 number. The common form of an exponential series is:
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