Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or rise in a specific base. For instance, let us suppose a country's population doubles yearly. This population growth can be represented in the form of an exponential function.
Exponential functions have numerous real-world use cases. Mathematically speaking, 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 is the equation 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, 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 a situation where b is higher than 0 and unequal to 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we have to find the spots where the function intersects the axes. This is referred to as 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 rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this approach, we determine the range values and the domain for the function. Once we have the rate, we need to chart them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar properties. When the base of an exponential function is greater than 1, the graph would 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 on an incline
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The graph is smooth and ongoing
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As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x advances toward positive infinity, the graph grows without bound.
In cases where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following qualities:
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The graph crosses 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 descending
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The graph is a curved line
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As x nears 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 constant
Rules
There are a few essential rules to bear in mind when dealing 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 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 an identical base, deduct the exponents.
For instance, if we need to divide two exponential functions that posses 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 example, if we have to raise 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 forever equal to 1.
For example, 1^x = 1 regardless of what the value 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 used to signify exponential growth. As the variable increases, the value of the function increases at a ever-increasing pace.
Example 1
Let’s observe the example of the growing of bacteria. If we have a cluster of bacteria that duplicates each hour, then at the close of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x 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 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
Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive material that degenerates at a rate of half its amount every hour, then at the end of one hour, we will have half as much material.
After the second hour, we will have a quarter as much material (1/2 x 1/2).
After the third hour, we will have one-eighth as much material (1/2 x 1/2 x 1/2).
This can be shown using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the quantity of material at time t and t is assessed in hours.
As shown, both of these samples use a comparable pattern, which is the reason they are able to be shown using exponential functions.
As a matter of fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base remains constant. This indicates that any exponential growth or decay where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate remains the same whilst the base is static in regular intervals of time.
Solution
An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we need to enter different values for x and then asses the corresponding values for y.
Let's look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As demonstrated, the values of y grow very fast as x grows. Imagine 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 rises from left to right ,getting steeper as it persists.
Example 2
Chart the following exponential function:
y = 1/2^x
First, let's create a table of values.
As you can see, the values of y decrease very quickly as x increases. 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 would look like this:
This is a decay function. As shown, the graph is a curved line that descends 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 written as f(ax)/dx = ax. All derivatives of exponential functions display special properties 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 general form of an exponential series is:
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