Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or increase in a certain base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be depicted as an exponential function.
Exponential functions have many real-world applications. Expressed mathematically, an exponential function is shown as f(x) = b^x.
In this piece, we will learn the basics of an exponential function in conjunction with important examples.
What’s the formula for an Exponential Function?
The common 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 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 greater than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we have to locate the dots where the function intersects the axes. This is called 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 focus on the value of b = 2.
To locate the y-coordinates, one must to set the worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
By following this method, we get the range values and the domain for the function. Once we determine the values, 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 qualities. When the base of an exponential function is greater than 1, the graph would have the following properties:
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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 more 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 continuous
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As x approaches negative infinity, the graph is asymptomatic towards the x-axis
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As x approaches positive infinity, the graph increases without bound.
In situations where the bases are fractions or decimals between 0 and 1, an exponential function displays the following properties:
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The graph passes the point (0,1)
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The range is greater than 0
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The domain is all 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 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 level
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The graph is constant
Rules
There are some vital rules to remember when working 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 that have 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 identical base, deduct the exponents.
For example, if we have 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 increase an exponential function to a power, multiply the exponents.
For instance, if we have to raise 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 with a base of 1 is forever equivalent 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 equivalent to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are commonly used to indicate exponential growth. As the variable rises, the value of the function rises quicker and quicker.
Example 1
Let's look at the example of the growth of bacteria. If we have a group of bacteria that duplicates every hour, then at the end of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have quadruple 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 an exponential function as follows:
f(t) = 2^t
where f(t) is the amount of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can portray exponential decay. If we have a dangerous substance that decomposes at a rate of half its amount every hour, then at the end of one hour, we will have half as much material.
At the end of two hours, we will have a quarter 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 substance at time t and t is calculated in hours.
As you can see, both of these samples pursue a comparable pattern, which is the reason they can be shown using exponential functions.
In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base continues to be fixed. This means that any exponential growth or decomposition where the base varies is not an exponential function.
For example, in the case of compound interest, the interest rate remains the same while the base changes in normal 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 input different values for x and then measure the matching values for y.
Let's 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 demonstrated, the worth of y increase very quickly as x grows. Consider 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 persists.
Example 2
Graph 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 rapidly as x surges. 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 the following:
The above is a decay function. As shown, the graph is a curved line that gets lower from right to left and gets smoother as it goes.
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 present particular properties 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:
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