Exponential Functions - Formula, Properties, Graph, Rules
What’s 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 yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have many real-life use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Here we will learn the essentials of an exponential function in conjunction with important examples.
What’s the equation for an Exponential Function?
The general 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 fixed, 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 larger than 0 and not equal to 1, x will be a real number.
How do you graph Exponential Functions?
To chart an exponential function, we must discover the spots where the function crosses the axes. These are referred to as the x and y-intercepts.
Considering the fact that the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To locate the y-coordinates, one must to set the value for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
According to this method, we get the domain and the range values for the function. Once we determine 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 qualities. When the base of an exponential function is larger than 1, the graph will have the following qualities:
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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 increasing
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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 grows without bound.
In situations where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following characteristics:
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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 descending
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The graph is a curved line
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As x approaches 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 level
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The graph is unending
Rules
There are a few basic rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with an identical 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 equivalent base, subtract the exponents.
For example, if we have to divide two exponential functions that have a base of 3, we can write it as 3^x / 3^y = 3^(x-y).
Rule 3: To grow 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 write 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 rate of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are usually utilized to denote exponential growth. As the variable rises, the value of the function grows faster and faster.
Example 1
Let's look at the example of the growth of bacteria. If we have a culture of bacteria that duplicates each hour, then at the close of hour one, 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 portrayed an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can illustrate exponential decay. If we have a radioactive substance 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 hour two, we will have one-fourth 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 represented using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the volume of material at time t and t is assessed in hours.
As you can see, both of these illustrations use 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 demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base stays constant. This indicates 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 stays the same whereas the base varies in regular intervals of time.
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 plug in different values for x and then asses the matching values for y.
Let's check out the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As demonstrated, the rates of y rise very rapidly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As shown, the graph is a curved line that rises from left to right and gets steeper as it continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To start, let's make a table of values.
As you can see, the values of y decrease very rapidly as x increases. The reason is because 1/2 is less than 1.
Let’s say we were to plot the x-values and y-values on a coordinate plane, it is going to look like what you see below:
This is a decay function. As you can see, the graph is a curved line that descends from right to left and gets flatter 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 exhibit particular features by which 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 expressions are the powers of an independent variable number. The general form of an exponential series is:
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