Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. For instance, let's say a country's population doubles annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-life applications. Mathematically speaking, an exponential function is written as f(x) = b^x.
In this piece, we discuss the essentials of an exponential function coupled with important 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 fixed, and x is a variable
For example, 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 the event where b is larger than 0 and does not equal 1, x will be a real number.
How do you graph Exponential Functions?
To graph an exponential function, we need to find the dots where the function intersects the axes. This is called the x and y-intercepts.
As the exponential function has a constant, we need to set the value for it. Let's focus on the value of b = 2.
To find the y-coordinates, we need 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 method, we determine the range values and the domain for the function. Once we determine 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 identical properties. When the base of an exponential function is greater than 1, the graph will 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 larger 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 constant
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As x nears negative infinity, the graph is asymptomatic towards 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 within 0 and 1, an exponential function displays the following properties:
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The graph intersects the point (0,1)
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The range is more than 0
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The domain is all 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 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 constant
Rules
There are a few basic rules to remember when engaging with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For example, if we have 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 an equivalent base, deduct the exponents.
For instance, if we need 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 raise 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 note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is consistently equal to 1.
For example, 1^x = 1 regardless of what the rate of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For instance, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly used to signify exponential growth. As the variable increases, the value of the function increases faster and faster.
Example 1
Let’s observe the example of the growing of bacteria. If we have a cluster of bacteria that multiples by two hourly, then at the end of hour one, we will have twice as many bacteria.
At the end of hour two, we will have 4x as many bacteria (2 x 2).
At the end of the third hour, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be displayed utilizing 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 dangerous 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.
At the end of hour two, we will have one-fourth as much substance (1/2 x 1/2).
After the third hour, we will have an eighth as much material (1/2 x 1/2 x 1/2).
This can be displayed using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the quantity of substance at time t and t is assessed in hours.
As demonstrated, both of these illustrations follow a comparable pattern, which is why they can be shown 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 denoted by the variable whereas the base continues to be fixed. This means that any exponential growth or decay where the base is different 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 utilizing a table of values. To get the graph of an exponential function, we have to plug in different values for x and calculate the equivalent values for y.
Let us look at the example below.
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 increase very rapidly as x grows. Consider 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 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 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.
If 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 shown, the graph is a curved line that descends 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 exhibit 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 figure. The common form of an exponential series is:
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