Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a certain base. Take this, for example, let's say a country's population doubles annually. This population growth can be depicted as an exponential function.
Exponential functions have numerous real-life use cases. Mathematically speaking, an exponential function is written as f(x) = b^x.
Today we discuss the essentials of an exponential function coupled with relevant 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 a constant, and x is a variable
For instance, 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 cases where b is greater than 0 and not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To chart an exponential function, we need to find the dots where the function intersects the axes. These are known as the x and y-intercepts.
As 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, 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 technique, we determine the domain and the range values for the function. Once we have the values, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable qualities. When the base of an exponential function is greater than 1, the graph is going to have the following 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 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 level and ongoing
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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 increases without bound.
In instances where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following attributes:
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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 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 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 flat
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The graph is constant
Rules
There are some essential rules to recall 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 with 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 equivalent base, subtract 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 raise 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 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 identical to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly used to signify exponential growth. As the variable grows, the value of the function increases at a ever-increasing pace.
Example 1
Let’s examine the example of the growth of bacteria. Let’s say we have a group of bacteria that multiples by two every hour, then at the end of the first hour, we will have double 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 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 represent exponential decay. If we have a dangerous material that decomposes at a rate of half its quantity every hour, then at the end of hour one, we will have half as much material.
After hour two, we will have 1/4 as much substance (1/2 x 1/2).
At the end of 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 below:
f(t) = 1/2^t
where f(t) is the amount of substance at time t and t is measured in hours.
As you can see, both of these samples use a similar pattern, which is the reason they can be depicted 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 represented by the variable while the base stays fixed. This means that any exponential growth or decomposition where the base varies is not an exponential function.
For example, in the matter of compound interest, the interest rate remains the same whereas the base changes 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 must enter different values for x and measure the equivalent values for y.
Let's 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 shown, the worth of y grow very quickly 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 goes.
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 swiftly as x rises. This is because 1/2 is less than 1.
If we were to chart the x-values and y-values on a coordinate plane, it would look like this:
The above 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions display special features whereby 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 number. The common form of an exponential series is:
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