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's say a country's population doubles annually. This population growth can be portrayed in the form of an exponential function.
Exponential functions have many real-world uses. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
Here we will learn the fundamentals of an exponential function along with relevant 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 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 a situation where b is greater than 0 and not equal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we need to find the spots where the function crosses the axes. This is referred to 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 discover the y-coordinates, we need to set the worth for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this technique, we achieve the range values and the domain for the function. After having the rate, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable properties. When the base of an exponential function is more than 1, the graph will have the following qualities:
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The line crosses 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 ongoing
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As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x nears positive infinity, the graph rises without bound.
In instances where the bases are fractions or decimals in the middle of 0 and 1, an exponential function exhibits the following properties:
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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 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 need to multiply two exponential functions that have 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, subtract the exponents.
For instance, if we have to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise 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 with a base of 1 is consistently equal to 1.
For instance, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly used to indicate exponential growth. As the variable grows, the value of the function increases at a ever-increasing pace.
Example 1
Let's look at the example of the growing of bacteria. If we have a cluster of bacteria that multiples by two every hour, then at the end of hour one, we will have double as many bacteria.
At the end of hour two, we will have 4 times 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 represented 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 in hours.
Example 2
Also, exponential functions can illustrate exponential decay. Let’s say we had a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of one hour, we will have half as much material.
After two hours, we will have one-fourth as much material (1/2 x 1/2).
After three hours, 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 below:
f(t) = 1/2^t
where f(t) is the amount of substance at time t and t is measured in hours.
As you can see, both of these illustrations follow a comparable pattern, which is why they can be represented using exponential functions.
In fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base stays fixed. This indicates 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 whilst the base varies 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 must input different values for x and asses the matching values for y.
Let's review this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the worth of y grow very rapidly as x increases. Imagine 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 goes.
Example 2
Graph the following exponential function:
y = 1/2^x
To begin, let's draw up a table of values.
As you can see, the values of y decrease very quickly as x increases. The reason is because 1/2 is less than 1.
Let’s say we were to plot the x-values and y-values on a coordinate plane, it would look like the following:
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 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 special properties whereby 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 digit. The general form of an exponential series is:
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