Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a particular base. For example, let us assume a country's population doubles every year. This population growth can be depicted in the form of an exponential function.
Exponential functions have multiple real-world uses. Mathematically speaking, an exponential function is written as f(x) = b^x.
Today we discuss the fundamentals of an exponential function in conjunction with appropriate examples.
What is the equation for an Exponential Function?
The general 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 varies
As an illustration, 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 a situation where b is larger than 0 and unequal to 1, x will be a real number.
How do you graph Exponential Functions?
To chart an exponential function, we must locate the points where the function intersects the axes. These are known as the x and y-intercepts.
As 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 find the y-coordinates, its essential to set the value for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
In following this approach, we achieve the domain and the range values for the function. Once we determine 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 identical characteristics. When the base of an exponential function is more than 1, the graph is going to have the following qualities:
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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 smooth 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 rises without bound.
In events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following attributes:
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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 all real numbers
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The graph is declining
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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 approaches negative infinity, the line approaches without bound
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The graph is smooth
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The graph is continuous
Rules
There are several essential rules to recall when engaging 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 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 equivalent base, subtract the exponents.
For instance, if we need to divide two exponential functions with 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 grow 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 that has a base of 1 is always equivalent to 1.
For instance, 1^x = 1 no matter what the worth of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually used to signify exponential growth. As the variable grows, the value of the function rises faster and faster.
Example 1
Let’s observe the example of the growth of bacteria. Let us suppose that we have a culture of bacteria that doubles each hour, then at the close of the first hour, we will have 2 times as many bacteria.
At the end of hour two, we will have 4 times 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 illustrate 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 substance.
At the end of the second hour, we will have 1/4 as much substance (1/2 x 1/2).
At the end of hour three, 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 below:
f(t) = 1/2^t
where f(t) is the quantity of material at time t and t is measured in hours.
As shown, both of these illustrations follow a comparable pattern, which is why they can be shown using exponential functions.
As a matter of fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is represented by the variable while the base continues to be the same. Therefore any exponential growth or decay where the base changes is not an exponential function.
For example, in the scenario of compound interest, the interest rate stays the same whereas the base changes in normal intervals of time.
Solution
An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we have 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
First, let's make a table of values.
As you can see, the worth of y increase very rapidly as x grows. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like this:
As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it persists.
Example 2
Draw the following exponential function:
y = 1/2^x
To start, let's create 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 is going to look like what you see below:
The above is a decay function. As shown, the graph is a curved line that descends from right to left and gets flatter as it continues.
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 special features by which 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 terms are the powers of an independent variable figure. The general form of an exponential series is:
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