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 as an exponential function.
Exponential functions have many real-world applications. 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 formula for an Exponential Function?
The common 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 fixed, and x varies
For example, 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 cases 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 crosses the axes. These are referred to as the x and y-intercepts.
Considering the fact that 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, its essential to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
According to this method, we get the range values and the domain for the function. Once we have the worth, we need to plot 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 larger than 1, the graph is going to 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 increasing
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The graph is level and continuous
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As x approaches negative infinity, the graph is asymptomatic regarding the x-axis
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As x nears positive infinity, the graph increases without bound.
In instances where the bases are fractions or decimals in the middle of 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 more 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 advances toward positive infinity, the line within 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 flat
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The graph is unending
Rules
There are some essential rules to bear in mind when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For instance, if we have to multiply two exponential functions that have 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 identical base, subtract the exponents.
For instance, if we have to divide two exponential functions that have a base of 3, we can write it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase an exponential function to a power, multiply the exponents.
For instance, 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 that has a base of 1 is always equal to 1.
For example, 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 no matter what the value of x is.
Examples
Exponential functions are usually utilized to signify exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.
Example 1
Let's look at the example of the growth of bacteria. Let’s say we have a culture of bacteria that doubles each 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 quadruple 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 an exponential function as follows:
f(t) = 2^t
where f(t) is the amount of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can represent exponential decay. If we have a dangerous substance that decomposes at a rate of half its volume every hour, then at the end of hour one, we will have half as much substance.
At the end of two hours, we will have a quarter as much substance (1/2 x 1/2).
At the end of three hours, we will have one-eighth as much material (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 volume of material at time t and t is measured in hours.
As you can see, both of these samples pursue a similar pattern, which is why they can be shown 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 whereas the base stays the same. This means that any exponential growth or decomposition where the base changes is not an exponential function.
For example, in the matter of compound interest, the interest rate stays the same while the base varies in ordinary time periods.
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 then measure the equivalent values for y.
Let us check out the example below.
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 values of y rise very rapidly as x rises. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like this:
As shown, the graph is a curved line that goes up from left to right and gets steeper as it continues.
Example 2
Chart the following exponential function:
y = 1/2^x
To start, let's create a table of values.
As shown, the values of y decrease very swiftly as x increases. This is because 1/2 is less than 1.
If we were to draw the x-values and y-values on a coordinate plane, it would look like the following:
This is a decay function. As shown, 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 particular 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 terms are the powers of an independent variable digit. The common form of an exponential series is:
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