Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a specific base. For example, let us assume a country's population doubles annually. This population growth can be portrayed as an exponential function.
Exponential functions have numerous real-world applications. Mathematically speaking, an exponential function is shown as f(x) = b^x.
In this piece, we will learn the essentials of an exponential function in conjunction with appropriate examples.
What is the formula for an Exponential Function?
The general 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 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 higher 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 must locate the points where the function intersects the axes. These are 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 locate the y-coordinates, we need to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
By following this approach, we achieve the domain and the range values for the function. Once we have 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 comparable characteristics. When the base of an exponential function is more than 1, the graph is going to have the below 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 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 flat and constant
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As x advances toward negative infinity, the graph is asymptomatic towards the x-axis
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As x approaches positive infinity, the graph grows without bound.
In situations where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following qualities:
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The graph crosses 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 declining
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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 approaches 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 essential rules to recall when engaging 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 compose 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 compose 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 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 that has a base of 1 is forever 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 no matter what the value of x is.
Examples
Exponential functions are generally leveraged to denote exponential growth. As the variable increases, the value of the function grows quicker and quicker.
Example 1
Let’s examine the example of the growth of bacteria. Let us suppose that we have a group of bacteria that multiples by two hourly, then at the close of hour one, we will have double as many bacteria.
At the end of the second hour, we will have 4 times 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 portrayed 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 portray exponential decay. Let’s say we had a dangerous material that decays 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 1/4 as much material (1/2 x 1/2).
At the end of three hours, we will have 1/8 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 quantity of substance at time t and t is assessed in hours.
As demonstrated, both of these illustrations follow a comparable pattern, which is the reason they can be shown using exponential functions.
In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base remains constant. Therefore any exponential growth or decay where the base changes is not an exponential function.
For instance, in the matter of compound interest, the interest rate continues to be the same whilst the base is static in ordinary time periods.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we need to enter different values for x and then calculate the corresponding values for y.
Let us check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the rates of y grow very fast 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 and gets steeper as it persists.
Example 2
Graph the following exponential function:
y = 1/2^x
First, let's make a table of values.
As you can see, the values of y decrease very quickly as x surges. This 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 flatter as it proceeds.
The Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x can be written as f(ax)/dx = ax. All derivatives of exponential functions display unique characteristics 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 figure. The common form of an exponential series is:
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