Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. For example, let us suppose a country's population doubles every year. This population growth can be represented as an exponential function.
Exponential functions have numerous real-life uses. In mathematical terms, an exponential function is written as f(x) = b^x.
In this piece, we discuss the fundamentals of an exponential function along with important examples.
What’s the equation for an Exponential Function?
The general 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 example, 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 chart Exponential Functions?
To chart an exponential function, we must discover the points where the function crosses the axes. This is referred to as the x and y-intercepts.
As the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, we need to set the value for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this method, we achieve the domain and the range values for the function. Once we have 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 similar qualities. When the base of an exponential function is greater than 1, the graph would have the below characteristics:
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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 on an incline
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The graph is flat and constant
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As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x advances toward 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 exhibits the following qualities:
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The graph crosses the point (0,1)
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The range is more 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 advances toward 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 some essential rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we need to multiply two exponential functions that posses 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 need to divide two exponential functions that have 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 increase 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 with a base of 1 is consistently equivalent to 1.
For example, 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 generally leveraged to signify exponential growth. As the variable rises, the value of the function increases quicker and quicker.
Example 1
Let’s examine the example of the growing of bacteria. Let us suppose that we have a cluster of bacteria that duplicates each hour, then at the close of the first hour, we will have twice as many bacteria.
At the end of hour two, we will have 4 times 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 represented an exponential function as follows:
f(t) = 2^t
where f(t) is the total sum of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can portray exponential decay. Let’s say we had a radioactive material that decays 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 two hours, we will have 1/4 as much substance (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 represented 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 measured in hours.
As you can see, both of these examples use a comparable pattern, which is why they can be depicted using exponential functions.
As a matter of fact, any rate of change can be indicated 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. This means that any exponential growth or decay where the base varies is not an exponential function.
For instance, in the case of compound interest, the interest rate continues to be the same whilst the base is static in normal amounts of time.
Solution
An exponential function can be graphed using a table of values. To get the graph of an exponential function, we need to input different values for x and asses the corresponding values for y.
Let's look at this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the worth of y rise very quickly as x rises. 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 goes.
Example 2
Graph the following exponential function:
y = 1/2^x
First, let's create a table of values.
As shown, the values of y decrease very rapidly as x rises. This 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 is going to look like this:
This 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 proceeds.
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 particular characteristics 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 figure. The common form of an exponential series is:
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