Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a specific base. Take this, for example, 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-life use cases. Expressed mathematically, an exponential function is written as f(x) = b^x.
Today we will review the essentials of an exponential function along with appropriate examples.
What’s the formula 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 is a variable
For instance, 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 a situation where b is higher than 0 and unequal to 1, x will be a real number.
How do you graph Exponential Functions?
To graph an exponential function, we need to discover the spots where the function crosses the axes. This is known as the x and y-intercepts.
Considering the fact that the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, one must to set the worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
In following this technique, we determine the domain and the range values for the function. After having the rate, we need to chart 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 larger than 1, the graph will have the below qualities:
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The line intersects 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 on an incline
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The graph is level and continuous
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As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x advances toward positive infinity, the graph grows without bound.
In cases where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following properties:
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The graph intersects the point (0,1)
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The range is larger 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 advances toward 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 a few basic rules to remember when engaging with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, 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 an equivalent base, deduct the exponents.
For example, if we need to divide two exponential functions that have 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 compose 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 no matter what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually used to denote exponential growth. As the variable increases, the value of the function grows quicker and quicker.
Example 1
Let's look at the example of the growth of bacteria. If we have a group of bacteria that doubles hourly, then at the close of hour one, we will have 2 times as many bacteria.
At the end of hour two, we will have quadruple 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 number of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a dangerous substance that decomposes at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
After the second hour, we will have 1/4 as much substance (1/2 x 1/2).
After three hours, we will have one-eighth as much material (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 volume of substance at time t and t is measured in hours.
As you can see, both of these examples follow a comparable pattern, which is why they can be represented using exponential functions.
In 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 represented by the variable whereas the base continues to be the same. This means that any exponential growth or decline where the base changes is not an exponential function.
For instance, in the scenario of compound interest, the interest rate remains the same whilst the base is static 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 need to input different values for x and then asses the equivalent values for y.
Let us check out the following 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 increase very fast as x increases. Consider we were to draw this exponential function graph on a coordinate plane, it would look like the following:
As shown, the graph is a curved line that rises from left to right and gets steeper as it goes.
Example 2
Chart the following exponential function:
y = 1/2^x
To begin, let's make a table of values.
As you can see, the values of y decrease very swiftly as x rises. The reason is because 1/2 is less than 1.
Let’s say we were to chart the x-values and y-values on a coordinate plane, it is going to look like this:
This 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 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 exhibit particular characteristics 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 figure. The general form of an exponential series is:
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