Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a specific base. For instance, let's say a country's population doubles annually. This population growth can be portrayed in the form of an exponential function.
Exponential functions have many real-world uses. Mathematically speaking, an exponential function is written as f(x) = b^x.
Here we will review the basics of an exponential function in conjunction with appropriate examples.
What is the formula for an Exponential Function?
The generic 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 is a variable
As an illustration, 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 greater than 0 and does not equal 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we must locate the dots where the function intersects the axes. This is called the x and y-intercepts.
As the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, its essential to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this approach, we get the range values and the domain for the function. Once we have the values, 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 qualities. When the base of an exponential function is greater than 1, the graph will have the following 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 more 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 level and constant
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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 grows without bound.
In cases where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following attributes:
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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 within 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 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 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 an equivalent base, subtract the exponents.
For instance, if we need to divide two exponential functions that have a base of 3, we can note 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, we are able to compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is consistently equal to 1.
For example, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are commonly utilized to signify exponential growth. As the variable increases, the value of the function grows at a ever-increasing pace.
Example 1
Let's look at the example of the growth of bacteria. Let’s say we have a cluster of bacteria that duplicates hourly, then at the end of the first hour, we will have double as many bacteria.
At the end of the second hour, we will have 4x 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 utilizing an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured in hours.
Example 2
Similarly, exponential functions can represent exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its amount every hour, then at the end of the first hour, we will have half as much substance.
At the end of the second hour, we will have 1/4 as much substance (1/2 x 1/2).
After three hours, we will have 1/8 as much substance (1/2 x 1/2 x 1/2).
This can be displayed 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 measured in hours.
As you can see, both of these samples pursue a similar pattern, which is the reason they are able to be represented using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base remains fixed. This means that any exponential growth or decline where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate remains the same while the base changes in regular amounts of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we need to enter different values for x and calculate the equivalent values for y.
Let's review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the rates of y increase very rapidly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As shown, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To begin, let's make a table of values.
As shown, the values of y decrease very swiftly as x rises. 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:
The above is a decay function. As shown, the graph is a curved line that gets lower 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions exhibit special properties where 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 terms are the powers of an independent variable number. The general form of an exponential series is:
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