Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. Take this, for example, let us suppose a country's population doubles every year. This population growth can be depicted as an exponential function.
Exponential functions have numerous real-world uses. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
Today we discuss the basics of an exponential function along with appropriate examples.
What is the formula 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
For instance, 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 unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we must locate the spots where the function crosses the axes. This is known as the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.
To discover 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 technique, we achieve the range values and the domain for the function. Once we determine the worth, we need to draw 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 greater than 1, the graph will have the below properties:
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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 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 advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x nears positive infinity, the graph increases without bound.
In instances where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following properties:
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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 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 advances toward 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 constant
Rules
There are several vital rules to recall when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we have to multiply two exponential functions that have 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, deduct 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 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 with a base of 1 is always equal to 1.
For instance, 1^x = 1 no matter what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are generally used to denote exponential growth. As the variable grows, the value of the function rises at a ever-increasing pace.
Example 1
Let's look at the example of the growth of bacteria. If we have a group of bacteria that multiples by two every hour, then at the end of hour one, we will have twice as many bacteria.
At the end of hour two, we will have quadruple 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 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
Also, exponential functions can represent exponential decay. If we have a radioactive material that decays at a rate of half its volume every hour, then at the end of the first hour, we will have half as much material.
After two hours, we will have a quarter as much material (1/2 x 1/2).
After three hours, 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 assessed in hours.
As shown, both of these illustrations follow a comparable pattern, which is why they are able to be represented using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base stays constant. This indicates that any exponential growth or decomposition where the base is different is not an exponential function.
For example, in the matter of compound interest, the interest rate continues to be the same whereas the base changes in normal time periods.
Solution
An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we need to enter different values for x and then asses the equivalent values for y.
Let's check out this example.
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 rise very quickly as x increases. Imagine 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 goes up from left to right and gets steeper as it continues.
Example 2
Plot the following exponential function:
y = 1/2^x
To begin, let's draw up a table of values.
As shown, the values of y decrease very swiftly as x increases. This 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 what you see below:
The above is a decay function. As shown, the graph is a curved line that decreases from right to left and gets smoother as it proceeds.
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 exhibit special properties whereby 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 digit. The general form of an exponential series is:
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