Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a certain base. For example, let us suppose a country's population doubles yearly. This population growth can be represented as an exponential function.
Exponential functions have many real-world uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we discuss the basics of an exponential function coupled with important examples.
What is the equation for an Exponential Function?
The generic 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
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 larger than 0 and does not equal 1, x will be a real number.
How do you chart Exponential Functions?
To chart an exponential function, we have to locate the spots where the function intersects the axes. This is referred to as the x and y-intercepts.
Since 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 locate the y-coordinates, its essential to set the rate for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
By following this technique, we get the range values and the domain 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 identical properties. When the base of an exponential function is greater 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 greater 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 level 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 nears positive infinity, the graph increases without bound.
In events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function exhibits the following properties:
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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 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 gets closer to negative infinity, the line approaches without bound
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The graph is level
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The graph is constant
Rules
There are a few basic rules to recall when dealing 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 with a base of 2, then we can note 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 write it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase an exponential function to a power, multiply the exponents.
For instance, if we have to increase an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is always 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 instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are generally utilized to signify exponential growth. As the variable grows, the value of the function grows faster and faster.
Example 1
Let's look at the example of the growing of bacteria. Let’s say we have a cluster of bacteria that doubles every hour, then at the end of hour one, we will have double 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 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed using an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured in hours.
Example 2
Similarly, exponential functions can represent exponential decay. If we have a dangerous material that decomposes at a rate of half its amount every hour, then at the end of one hour, we will have half as much material.
After two hours, we will have a quarter 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 displayed using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the amount of substance at time t and t is calculated in hours.
As shown, both of these samples pursue a similar pattern, which is why they can be represented using exponential functions.
As a matter of fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base remains the same. This indicates that any exponential growth or decomposition where the base is different is not an exponential function.
For instance, in the matter of compound interest, the interest rate stays the same whereas the base varies in normal intervals of time.
Solution
An exponential function can 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 corresponding values for y.
Let us look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As you can see, the rates of y rise very fast as x grows. If we were to plot 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 persists.
Example 2
Draw the following exponential function:
y = 1/2^x
To start, let's make a table of values.
As you can see, the values of y decrease very quickly as x surges. 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 would look like what you see below:
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 flatter as it proceeds.
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 unique 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 terms are the powers of an independent variable digit. The general form of an exponential series is:
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