Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. For example, let us assume a country's population doubles yearly. This population growth can be represented in the form of an exponential function.
Exponential functions have many real-life applications. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
In this piece, we will review the fundamentals of an exponential function coupled with appropriate examples.
What is 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 fixed, and x is a variable
For instance, 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 not equal to 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we must find the points where the function crosses the axes. This is called the x and y-intercepts.
Considering the fact that the exponential function has a constant, it will be necessary to set the value for it. Let's take the value of b = 2.
To discover the y-coordinates, we need 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 technique, we determine the domain and the range values for the function. After having 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 properties. When the base of an exponential function is larger than 1, the graph will have the below 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 greater 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 constant
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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 instances where the bases are fractions or decimals between 0 and 1, an exponential function displays the following qualities:
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The graph intersects 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 descending
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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 level
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The graph is constant
Rules
There are several essential rules to remember when dealing 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 have 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 equivalent base, deduct 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 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, we are able to write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is forever equal to 1.
For instance, 1^x = 1 regardless of what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For example, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually utilized to denote exponential growth. As the variable grows, the value of the function grows quicker and quicker.
Example 1
Let’s observe the example of the growth of bacteria. Let’s say we have a group of bacteria that duplicates every hour, 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 hour three, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be displayed using 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
Also, exponential functions can illustrate exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its volume every hour, then at the end of the first hour, we will have half as much material.
At the end of the second hour, we will have 1/4 as much material (1/2 x 1/2).
After three hours, 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 amount of substance at time t and t is measured in hours.
As demonstrated, both of these illustrations use a similar pattern, which is the reason they can be depicted using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base stays the same. Therefore any exponential growth or decline where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate continues to be the same whilst the base varies in normal amounts of time.
Solution
An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we must plug in different values for x and then measure the matching values for y.
Let's review the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As you can see, the rates of y rise very quickly as x increases. If we were to draw 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 and gets steeper as it persists.
Example 2
Chart 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 surges. This 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 the following:
The above is a decay function. As you can see, the graph is a curved line that decreases from right to left and gets smoother 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 display special properties whereby 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 common form of an exponential series is:
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