Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a certain base. For example, 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-world use cases. Mathematically speaking, an exponential function is written as f(x) = b^x.
Here we will learn the fundamentals of an exponential function coupled with appropriate 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 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 plot Exponential Functions?
To graph an exponential function, we must locate the dots where the function crosses the axes. This is referred to as the x and y-intercepts.
As the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.
To find the y-coordinates, its essential 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 method, we achieve 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 similar qualities. When the base of an exponential function is greater than 1, the graph will 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 larger than 0
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The graph is a curved line
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The graph is increasing
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The graph is smooth and continuous
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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 increases without bound.
In cases where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following qualities:
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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 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 approaches 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 smooth
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The graph is unending
Rules
There are several basic rules to bear in mind when engaging 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 compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, deduct the exponents.
For instance, 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 grow an exponential function to a power, multiply the exponents.
For example, 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 that has a base of 1 is forever equal 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 example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly utilized to denote exponential growth. As the variable grows, the value of the function grows faster and faster.
Example 1
Let’s examine the example of the growing of bacteria. Let’s say we have a culture of bacteria that doubles each hour, then at the close 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 hour three, we will have 8x as many bacteria (2 x 2 x 2).
This rate of growth can be represented utilizing an exponential function as follows:
f(t) = 2^t
where f(t) is the total sum of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can illustrate exponential decay. If we have a dangerous material 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 material (1/2 x 1/2).
After 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 volume 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 depicted using exponential functions.
In fact, any rate of change can be demonstrated 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 constant. This means that any exponential growth or decline where the base is different is not an exponential function.
For example, in the scenario of compound interest, the interest rate stays the same while the base changes in regular intervals of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to enter different values for x and calculate the equivalent values for y.
Let's check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As demonstrated, the values of y rise very rapidly as x grows. If we were to draw this exponential function graph on a coordinate plane, it would look like the following:
As you can see, the graph is a curved line that goes up from left to right ,getting steeper as it continues.
Example 2
Draw the following exponential function:
y = 1/2^x
First, let's make a table of values.
As shown, the values of y decrease very rapidly as x rises. 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 is going to look like the following:
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 continues.
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 display unique properties by which 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 expressions are the powers of an independent variable digit. The common form of an exponential series is:
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