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 us assume a country's population doubles annually. This population growth can be represented in the form of an exponential function.
Exponential functions have many real-life applications. Expressed mathematically, an exponential function is shown as f(x) = b^x.
Here we will learn the basics of an exponential function along with appropriate examples.
What’s the equation for an Exponential Function?
The generic 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 varies
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 larger than 0 and not equal to 1, x will be a real number.
How do you graph Exponential Functions?
To chart an exponential function, we must discover the spots where the function crosses the axes. This is referred to 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 take the value of b = 2.
To locate the y-coordinates, its essential to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this approach, we determine the range values and the domain for the function. After having the rate, 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 characteristics. When the base of an exponential function is larger than 1, the graph will have the below properties:
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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 approaches 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 displays the following characteristics:
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The graph crosses the point (0,1)
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The range is larger than 0
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The domain is entirely real numbers
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The graph is descending
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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 several basic rules to recall when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we need 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, subtract the exponents.
For instance, if we need 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 grow 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, we are able to note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is consistently 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 instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are commonly leveraged to indicate exponential growth. As the variable increases, the value of the function grows faster and faster.
Example 1
Let’s examine the example of the growth of bacteria. Let’s say we have a cluster of bacteria that multiples by two hourly, then at the close of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have quadruple 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 portrayed using 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
Similarly, exponential functions can illustrate exponential decay. Let’s say we had a radioactive substance that decays at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
At the end of two hours, we will have 1/4 as much substance (1/2 x 1/2).
At the end of the third hour, 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 below:
f(t) = 1/2^t
where f(t) is the volume of substance at time t and t is assessed in hours.
As shown, both of these samples pursue a similar pattern, which is why they can be represented 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 remains fixed. This means that any exponential growth or decay where the base is different is not an exponential function.
For example, in the case of compound interest, the interest rate stays the same while the base is static in regular amounts of time.
Solution
An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we must plug in different values for x and measure the equivalent values for y.
Let's review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As demonstrated, the rates of y increase very quickly as x rises. Consider we were to plot this exponential function graph on a coordinate plane, it would look like this:
As shown, the graph is a curved line that goes up from left to right and gets steeper as it continues.
Example 2
Chart the following exponential function:
y = 1/2^x
First, let's make a table of values.
As shown, the values of y decrease very quickly as x increases. This is because 1/2 is less than 1.
If we were to plot the x-values and y-values on a coordinate plane, it is going to 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 flatter as it continues.
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 present 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 expressions are the powers of an independent variable figure. The common form of an exponential series is:
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