Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. For example, let us assume a country's population doubles every year. This population growth can be portrayed as an exponential function.
Exponential functions have many real-life use cases. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
In this piece, we will review the essentials of an exponential function along with relevant examples.
What’s 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
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 unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we need to locate 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, one must set the value for it. Let's take 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 method, we get the range values and the domain for the function. Once we determine the rate, 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 characteristics. When the base of an exponential function is more than 1, the graph would have the following qualities:
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The line intersects 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 level and ongoing
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As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
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As x approaches positive infinity, the graph grows without bound.
In cases 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 crosses 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 approaches positive infinity, the line in the 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 continuous
Rules
There are several vital rules to recall when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent 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 the same base, subtract the exponents.
For instance, if we have 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 grow 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 compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is consistently equivalent 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 equivalent to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are usually leveraged to signify exponential growth. As the variable rises, the value of the function grows at a ever-increasing pace.
Example 1
Let's look at the example of the growing of bacteria. Let’s say we have a culture of bacteria that duplicates hourly, then at the close of hour one, 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 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 number of bacteria at time t and t is measured hourly.
Example 2
Similarly, exponential functions can illustrate exponential decay. If we have a radioactive material that decays at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.
After two hours, we will have 1/4 as much material (1/2 x 1/2).
After the third hour, we will have 1/8 as much material (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 quantity of material at time t and t is calculated in hours.
As demonstrated, both of these examples pursue a similar pattern, which is why they are able to be shown 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 while the base stays fixed. This means that any exponential growth or decline where the base changes is not an exponential function.
For instance, in the matter of compound interest, the interest rate remains the same whereas the base is static in regular intervals of time.
Solution
An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we must input different values for x and then calculate the corresponding values for y.
Let's review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As you can see, the rates of y increase very fast as x grows. If we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Chart the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As you can see, the values of y decrease very swiftly as x surges. The reason is because 1/2 is less than 1.
Let’s say we were to chart the x-values and y-values on a coordinate plane, it would look like the following:
The above is a decay function. As shown, the graph is a curved line that decreases 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 written as f(ax)/dx = ax. All derivatives of exponential functions present particular features whereby 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 digit. The general form of an exponential series is:
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