Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or increase in a certain base. For instance, let us assume a country's population doubles annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have multiple real-world applications. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
Here we discuss the fundamentals of an exponential function in conjunction with relevant examples.
What is the equation 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 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 a situation where b is larger than 0 and unequal to 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we need to find the dots 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 discover the y-coordinates, one must to set the worth for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
In following this technique, we determine the range values and the domain for the function. Once we determine the values, we need to chart them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share identical qualities. When the base of an exponential function is greater 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 greater 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 level and ongoing
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As x advances toward negative infinity, the graph is asymptomatic towards the x-axis
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As x nears positive infinity, the graph grows without bound.
In situations where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following characteristics:
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The graph passes the point (0,1)
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The range is more than 0
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The domain is entirely real numbers
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The graph is descending
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The graph is a curved line
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As x nears positive infinity, the line in the 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 smooth
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The graph is constant
Rules
There are some basic rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we have to multiply two exponential functions with a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, deduct the exponents.
For example, if we have to divide two exponential functions with 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 grow an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is forever equivalent to 1.
For instance, 1^x = 1 no matter what the rate 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 commonly utilized to signify exponential growth. As the variable grows, the value of the function increases at a ever-increasing pace.
Example 1
Let’s examine the example of the growing of bacteria. If we have a culture of bacteria that doubles every hour, then at the end of hour one, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x as many bacteria (2 x 2).
At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can portray exponential decay. If we have a dangerous substance that decays at a rate of half its volume 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 substance (1/2 x 1/2).
After hour three, we will have one-eighth as much substance (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 substance at time t and t is assessed in hours.
As demonstrated, both of these illustrations pursue a comparable pattern, which is the reason they are able to be represented 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 while the base stays fixed. This means that any exponential growth or decomposition where the base varies is not an exponential function.
For instance, in the matter of compound interest, the interest rate remains the same while the base is static in regular intervals of time.
Solution
An exponential function can be graphed using a table of values. To get the graph of an exponential function, we need to enter different values for x and then calculate the matching values for y.
Let's check out the following example.
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 grow very fast as x rises. 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 begin, let's make a table of values.
As shown, the values of y decrease very quickly as x increases. 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 what you see below:
This is a decay function. As shown, the graph is a curved line that decreases from right to left and gets smoother as it goes.
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 particular characteristics 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 figure. The general form of an exponential series is:
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