Exponential Functions - Formula, Properties, Graph, Rules
What’s 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 annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-world applications. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Here we will review the fundamentals of an exponential function in conjunction with relevant examples.
What’s the equation for an Exponential Function?
The generic 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 unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we have to locate the spots where the function intersects the axes. These are called the x and y-intercepts.
Since 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 locate the y-coordinates, one must to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this method, we get the domain and the range values for the function. After having the rate, we need to graph them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share identical properties. When the base of an exponential function is larger than 1, the graph would have the following 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 larger than 0
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The graph is a curved line
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The graph is rising
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The graph is smooth and ongoing
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As x approaches negative infinity, the graph is asymptomatic towards the x-axis
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As x nears positive infinity, the graph rises without bound.
In cases where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following properties:
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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 all 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 advances toward 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 constant
Rules
There are a few basic rules to remember 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 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 have 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 example, if we have to increase an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is always equal to 1.
For example, 1^x = 1 no matter what the rate of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly used to signify exponential growth. As the variable grows, the value of the function increases quicker and quicker.
Example 1
Let's look at the example of the growth of bacteria. If we have a culture of bacteria that duplicates every hour, then at the close of hour one, we will have double 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 8x as many bacteria (2 x 2 x 2).
This rate of growth can be represented 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
Moreover, exponential functions can represent 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 one hour, we will have half as much substance.
At the end of hour two, we will have 1/4 as much substance (1/2 x 1/2).
After the third hour, we will have one-eighth as much material (1/2 x 1/2 x 1/2).
This can be displayed 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 continues to be constant. This means that any exponential growth or decomposition where the base changes is not an exponential function.
For instance, in the scenario of compound interest, the interest rate stays the same while the base varies in normal intervals of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to plug in different values for x and then measure the equivalent values for y.
Let us review the example below.
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 values of y grow very rapidly as x grows. Consider we were to draw this exponential function graph on a coordinate plane, it would look like this:
As you can see, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Chart the following exponential function:
y = 1/2^x
First, let's make a table of values.
As shown, 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 draw 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 gets lower 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 written as f(ax)/dx = ax. All derivatives of exponential functions exhibit special properties whereby 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 expressions are the powers of an independent variable figure. The common form of an exponential series is:
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