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 assume a country's population doubles every year. This population growth can be represented in the form of an exponential function.
Exponential functions have multiple real-world applications. In mathematical terms, an exponential function is displayed as f(x) = b^x.
In this piece, we discuss the essentials of an exponential function in conjunction with important examples.
What’s the formula for an Exponential Function?
The general 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 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 cases where b is higher than 0 and not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we must find the dots where the function crosses the axes. This is called the x and y-intercepts.
Considering the fact that the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.
To discover the y-coordinates, one must to set the value for x. For instance, 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 have 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 identical properties. When the base of an exponential function is greater 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 greater than 0
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The graph is a curved line
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The graph is rising
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The graph is flat and constant
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As x approaches negative infinity, the graph is asymptomatic towards 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 between 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 larger than 0
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The domain is all 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 approaches positive infinity, the line in the graph is asymptotic to the x-axis.
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As x advances toward negative infinity, the line approaches without bound
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The graph is level
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The graph is continuous
Rules
There are some vital rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we need to multiply two exponential functions that posses 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, deduct 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 increase 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 always equivalent to 1.
For example, 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 example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually used to indicate 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’s say we have a group of bacteria that doubles hourly, then at the end of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have quadruple as many bacteria (2 x 2).
At the end of hour three, we will have 8x 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 amount of bacteria at time t and t is measured hourly.
Example 2
Similarly, exponential functions can represent exponential decay. If we have a dangerous substance that degenerates at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
After hour two, we will have one-fourth as much material (1/2 x 1/2).
After the third hour, we will have 1/8 as much substance (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 amount of material at time t and t is assessed in hours.
As you can see, both of these examples use a similar pattern, which is why they can be shown using exponential functions.
As a matter of fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base continues to be constant. This means that any exponential growth or decline where the base is different is not an exponential function.
For example, in the case of compound interest, the interest rate remains the same whereas the base changes in normal amounts of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we must enter different values for x and then measure the corresponding values for y.
Let us 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 you can see, the worth of y rise very fast as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As you can see, the graph is a curved line that goes up from left to right and gets steeper as it goes.
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 quickly as x rises. The reason is because 1/2 is less than 1.
If we were to graph 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 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 written as f(ax)/dx = ax. All derivatives of exponential functions exhibit particular features whereby 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 terms are the powers of an independent variable digit. The common form of an exponential series is:
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