Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures 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 portrayed in the form of an exponential function.
Exponential functions have many real-life applications. In mathematical terms, an exponential function is shown as f(x) = b^x.
Today we discuss the essentials of an exponential function along with relevant examples.
What is 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 a constant, and x varies
For instance, 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 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 need to locate the dots where the function intersects the axes. These are referred to as the x and y-intercepts.
Since the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To locate 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.
By following this technique, we get the range values and the domain for the function. After having the worth, we need to draw them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar qualities. When the base of an exponential function is larger than 1, the graph would have the following properties:
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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 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 concerning the x-axis
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As x advances toward positive infinity, the graph rises without bound.
In situations where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following attributes:
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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 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 flat
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The graph is unending
Rules
There are some vital rules to bear in mind when engaging 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 posses a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, subtract the exponents.
For example, if we have 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 grow an exponential function to a power, multiply the exponents.
For example, if we have to raise 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 forever equivalent to 1.
For instance, 1^x = 1 regardless of what the rate 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 leveraged to indicate 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. If we have a culture of bacteria that doubles every hour, then at the end of the first hour, 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 hour three, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be represented 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
Also, exponential functions can portray exponential decay. Let’s say we had a radioactive substance that degenerates at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.
After hour two, we will have 1/4 as much material (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 below:
f(t) = 1/2^t
where f(t) is the volume of material at time t and t is assessed in hours.
As you can see, both of these examples pursue a similar pattern, which is why they are able to be shown using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Keep 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 indicates that any exponential growth or decomposition where the base is different is not an exponential function.
For example, in the case of compound interest, the interest rate continues to be the same while the base varies in regular 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 must input different values for x and then measure the corresponding values for y.
Let us check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As shown, the rates of y rise very rapidly as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like this:
As you can see, the graph is a curved line that rises from left to right and gets steeper as it continues.
Example 2
Chart the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As you can see, the values of y decrease very rapidly as x surges. The reason is because 1/2 is less than 1.
If we were to draw the x-values and y-values on a coordinate plane, it is going to look like this:
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 shown as f(ax)/dx = ax. All derivatives of exponential functions present unique features 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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