Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a certain base. For example, let us suppose a country's population doubles every year. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-life applications. Expressed mathematically, an exponential function is shown as f(x) = b^x.
Today we discuss the fundamentals of an exponential function coupled with relevant examples.
What’s the formula for an Exponential Function?
The common 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 varies
For instance, 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 the event where b is greater than 0 and not equal to 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we must discover the spots where the function crosses the axes. These are known as the x and y-intercepts.
Since 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, its essential to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this technique, we achieve the range values and the domain for the function. Once we determine the values, we need to graph 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 greater than 1, the graph would 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 increasing
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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 advances toward positive infinity, the graph increases without bound.
In events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function exhibits the following qualities:
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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 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 unending
Rules
There are some essential rules to remember when working with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For instance, if we need to multiply two exponential functions that posses a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, deduct the exponents.
For example, if we have to divide two exponential functions that posses 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 instance, if we have to grow an exponential function with a base of 4 to the third power, we are able to note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is consistently equal to 1.
For instance, 1^x = 1 no matter what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are commonly used to signify exponential growth. As the variable increases, the value of the function increases at a ever-increasing pace.
Example 1
Let's look at the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that doubles hourly, then at the end of the first hour, we will have twice as many bacteria.
At the end of hour two, we will have 4x 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 utilizing 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. If we have a radioactive material that decomposes at a rate of half its quantity every hour, then at the end of hour one, we will have half as much material.
After the second hour, we will have 1/4 as much material (1/2 x 1/2).
After the third hour, we will have one-eighth as much substance (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 volume of material at time t and t is calculated in hours.
As demonstrated, both of these illustrations use a comparable pattern, which is why they can be depicted using exponential functions.
In fact, any rate of change can be denoted using exponential functions. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable while the base continues to be constant. Therefore any exponential growth or decline where the base changes is not an exponential function.
For example, in the case of compound interest, the interest rate remains the same whilst the base is static in ordinary time periods.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we must plug in different values for x and calculate the corresponding values for y.
Let us review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the rates 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 this:
As shown, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Plot the following exponential function:
y = 1/2^x
First, let's draw up a table of values.
As you can see, the values of y decrease very quickly as x surges. The reason is because 1/2 is less than 1.
Let’s say we were to graph 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 you can see, the graph is a curved line that decreases 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions present special characteristics 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:
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