Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or increase in a certain base. Take this, for example, let us assume a country's population doubles annually. This population growth can be represented in the form of an exponential function.
Exponential functions have many real-life use cases. In mathematical terms, an exponential function is displayed as f(x) = b^x.
Here we will learn the basics of an exponential function in conjunction with relevant examples.
What is the equation 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 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 cases where b is higher than 0 and unequal to 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we need to find the points where the function crosses the axes. These are called the x and y-intercepts.
As the exponential function has a constant, one must 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 approach, we get the range values and the domain for the function. Once we have the worth, 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 more than 1, the graph will have the following characteristics:
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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 constant
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As x approaches negative infinity, the graph is asymptomatic concerning 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 within 0 and 1, an exponential function presents with the following qualities:
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The graph passes the point (0,1)
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The range is greater than 0
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The domain is entirely 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 level
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The graph is constant
Rules
There are some essential rules to recall when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, 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 the same base, deduct the exponents.
For instance, if we have to divide two exponential functions with 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 grow an exponential function with a base of 4 to the third power, we are able to compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with 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 example, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually used to indicate exponential growth. As the variable increases, the value of the function grows quicker and quicker.
Example 1
Let’s examine 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 hour one, we will have 2 times 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 portrayed 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 hourly.
Example 2
Similarly, exponential functions can portray exponential decay. If we have a dangerous substance that decays at a rate of half its quantity every hour, then at the end of one hour, we will have half as much substance.
At the end of the second hour, we will have one-fourth as much material (1/2 x 1/2).
After the third hour, we will have 1/8 as much substance (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 quantity of material at time t and t is assessed in hours.
As demonstrated, both of these illustrations use a similar pattern, which is why they can be depicted using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base continues to be fixed. This means that any exponential growth or decay 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 normal time periods.
Solution
An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we need to input different values for x and then calculate the matching values for y.
Let us look at 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 grows. Consider we were to plot 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 continues.
Example 2
Plot the following exponential function:
y = 1/2^x
First, let's create a table of values.
As shown, 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:
The above is a decay function. As shown, the graph is a curved line that gets lower 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions present particular characteristics 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 terms are the powers of an independent variable number. The common form of an exponential series is:
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