Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be portrayed as an exponential function.
Exponential functions have many real-world applications. In mathematical terms, an exponential function is written as f(x) = b^x.
Today we will review the fundamentals of an exponential function in conjunction with important examples.
What’s 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 is a variable
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 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 discover the points where the function crosses the axes. These are known as the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.
To find the y-coordinates, one must to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
In following this approach, we determine the range values and the domain for the function. After having the values, 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 characteristics. When the base of an exponential function is more than 1, the graph is going to have the following properties:
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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 rising
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The graph is smooth and continuous
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As x approaches negative infinity, the graph is asymptomatic towards the x-axis
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As x advances toward positive infinity, the graph increases without bound.
In instances 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 all 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 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 continuous
Rules
There are some basic rules to remember when working 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 posses a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, deduct the exponents.
For example, 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 grow 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 write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is consistently equivalent to 1.
For example, 1^x = 1 regardless of what the value of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually utilized to denote exponential growth. As the variable grows, the value of the function increases quicker and quicker.
Example 1
Let's look at the example of the growing of bacteria. Let us suppose that we have a cluster of bacteria that doubles each hour, then at the close 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 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
Similarly, 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 the first hour, we will have half as much material.
After two hours, we will have a quarter as much substance (1/2 x 1/2).
At the end of hour three, we will have 1/8 as much substance (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 quantity of material at time t and t is measured in hours.
As you can see, both of these illustrations pursue a comparable pattern, which is why they can be represented using exponential functions.
As a matter of fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base remains constant. This indicates that any exponential growth or decay where the base is different is not an exponential function.
For example, in the matter of compound interest, the interest rate continues to be the same while the base varies in regular time periods.
Solution
An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we have to input different values for x and calculate the matching values for y.
Let us review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the worth of y increase very rapidly as x rises. Imagine we were to draw 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 continues.
Example 2
Plot 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 rapidly as x rises. The reason is because 1/2 is less than 1.
If we were to plot 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 you can see, the graph is a curved line that gets lower 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions present unique properties by which 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 number. The general form of an exponential series is:
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