Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. For example, let us assume a country's population doubles every year. This population growth can be depicted in the form of an exponential function.
Exponential functions have many real-world applications. Mathematically speaking, an exponential function is displayed as f(x) = b^x.
Here we will review the essentials of an exponential function in conjunction with appropriate examples.
What’s the formula for an Exponential Function?
The general 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 varies
For instance, 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 chart an exponential function, we have to locate the dots where the function intersects the axes. This is referred to as the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.
To discover the y-coordinates, we need to set the value for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this technique, we get the domain and the range values for the function. Once we have the values, we need to chart 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 is going to have the following characteristics:
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The line passes 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 on an incline
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The graph is smooth and ongoing
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As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x approaches positive infinity, the graph grows without bound.
In events where the bases are fractions or decimals within 0 and 1, an exponential function exhibits 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 all real numbers
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The graph is decreasing
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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 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 continuous
Rules
There are some basic rules to bear in mind when dealing with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we have 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 an identical base, subtract the exponents.
For example, if we need to divide two exponential functions that have a base of 3, we can note 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 grow an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with a base of 1 is forever equivalent to 1.
For example, 1^x = 1 no matter what the rate 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 usually utilized to denote exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.
Example 1
Let’s observe the example of the growing of bacteria. Let’s say we have a group of bacteria that multiples by two hourly, then at the close of the first hour, 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 the third hour, 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 total sum of bacteria at time t and t is measured in hours.
Example 2
Similarly, exponential functions can portray exponential decay. If we have a radioactive material that decomposes at a rate of half its volume every hour, then at the end of the first hour, we will have half as much substance.
After hour two, we will have 1/4 as much substance (1/2 x 1/2).
After the third hour, we will have 1/8 as much material (1/2 x 1/2 x 1/2).
This can be represented using an exponential equation as below:
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 samples use a comparable pattern, which is the reason they are able to be shown using exponential functions.
As a matter of fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable while 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 scenario of compound interest, the interest rate remains the same while the base is static in ordinary time periods.
Solution
An exponential function is able to 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 asses the corresponding values for y.
Let's look at this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As demonstrated, the values of y increase very fast as x grows. Consider we were to draw this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that rises from left to right ,getting steeper as it continues.
Example 2
Graph 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 surges. This is because 1/2 is less than 1.
Let’s say we were to graph the x-values and y-values on a coordinate plane, it would look like what you see below:
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 goes.
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 present special characteristics where 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 terminology are the powers of an independent variable figure. The common form of an exponential series is:
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