Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or increase in a particular base. For example, let us suppose a country's population doubles yearly. This population growth can be represented as an exponential function.
Exponential functions have multiple real-world uses. Expressed mathematically, an exponential function is written as f(x) = b^x.
Today we will learn the basics of an exponential function along with important 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 a constant, and x varies
For example, 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 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 plot an exponential function, we must find the spots where the function crosses the axes. This is referred to as the x and y-intercepts.
Since the exponential function has a constant, it will be necessary to set the value for it. Let's take 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
By following this method, we determine the domain and the range values 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 characteristics. When the base of an exponential function is larger than 1, the graph is going to have the below 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 greater 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 advances toward 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 cases where the bases are fractions or decimals in the middle of 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 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 advances toward positive infinity, the line within 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 constant
Rules
There are some vital rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For example, if we need to multiply two exponential functions with a base of 2, then we can note 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 have 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 increase an exponential function to a power, multiply the exponents.
For instance, if we have to increase 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 consistently equal to 1.
For example, 1^x = 1 regardless of what the rate of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For example, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are generally leveraged to denote exponential growth. As the variable increases, the value of the function increases quicker and quicker.
Example 1
Let’s observe the example of the growth of bacteria. Let us suppose that we have a group of bacteria that duplicates every hour, then at the close of the first hour, we will have twice 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 portrayed 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
Similarly, exponential functions can represent exponential decay. Let’s say we had a radioactive material that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.
After hour two, we will have 1/4 as much material (1/2 x 1/2).
At the end of the third hour, we will have an 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 samples use 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. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base continues to be the same. Therefore any exponential growth or decay where the base is different is not an exponential function.
For instance, in the matter of compound interest, the interest rate continues to be the same whereas the base is static in normal time periods.
Solution
An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we have to input different values for x and then asses the equivalent values for y.
Let's look at this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the worth of y grow very fast as x increases. If 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 goes up from left to right ,getting steeper as it goes.
Example 2
Draw the following exponential function:
y = 1/2^x
To start, let's draw up a table of values.
As shown, the values of y decrease very rapidly as x rises. This is because 1/2 is less than 1.
If we were to chart 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 you can see, the graph is a curved line that descends from right to left and gets smoother as it continues.
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 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 terms are the powers of an independent variable number. The general form of an exponential series is:
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