Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. For example, let us suppose a country's population doubles annually. This population growth can be depicted as an exponential function.
Exponential functions have numerous real-life uses. Expressed mathematically, an exponential function is displayed as f(x) = b^x.
Here we will review the basics of an exponential function along with appropriate examples.
What’s the equation for an Exponential Function?
The generic 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 is a variable
As an illustration, 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 cases where b is larger than 0 and unequal to 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we have to find the points where the function crosses the axes. This is 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 discover the y-coordinates, we need to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
In following this approach, we get the range values and the domain for the function. Once we have 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 similar properties. When the base of an exponential function is greater than 1, the graph will 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 larger 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 level and continuous
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As x advances toward 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 within 0 and 1, an exponential function exhibits the following qualities:
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The graph passes 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 advances toward negative infinity, the line approaches without bound
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The graph is level
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The graph is continuous
Rules
There are a few vital rules to bear in mind when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For example, if we have 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 the same base, subtract the exponents.
For instance, if we need to divide two exponential functions that posses a base of 3, we can write it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For example, if we have to raise an exponential function with a base of 4 to the third power, then we can note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is always 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 equivalent to 0.
For example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are generally utilized to denote exponential growth. As the variable grows, the value of the function rises at a ever-increasing pace.
Example 1
Let’s examine the example of the growth of bacteria. Let’s say we have a group of bacteria that multiples by two every hour, then at the close of hour one, 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 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 total sum of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a radioactive material that degenerates at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.
At the end of the second hour, 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 represented using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the amount of material at time t and t is calculated in hours.
As demonstrated, both of these samples pursue 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 while the base stays the same. Therefore any exponential growth or decline where the base changes is not an exponential function.
For instance, in the case of compound interest, the interest rate remains the same whereas the base varies in normal time periods.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to enter different values for x and measure the equivalent values for y.
Let us look at this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As demonstrated, the worth of y grow very quickly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As you can see, the graph is a curved line that goes up from left to right and gets steeper as it persists.
Example 2
Draw 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 quickly as x rises. This is because 1/2 is less than 1.
If we were to chart the x-values and y-values on a coordinate plane, it would look like what you see below:
This is a decay function. As you can see, 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 written as f(ax)/dx = ax. All derivatives of exponential functions present unique features where 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 common form of an exponential series is:
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