Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. Take this, for example, let's say a country's population doubles annually. This population growth can be represented as an exponential function.
Exponential functions have numerous real-life use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
In this piece, we will learn the fundamentals of an exponential function coupled with relevant examples.
What is the formula for an Exponential Function?
The common 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 varies
For example, 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 graph Exponential Functions?
To chart an exponential function, we need to find the points where the function crosses the axes. These are known 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 focus on the value of b = 2.
To locate the y-coordinates, its essential to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
In following this method, we get the domain and the range values for the function. Once we have the worth, 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 characteristics. When the base of an exponential function is larger than 1, the graph will 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 flat and ongoing
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As x nears negative infinity, the graph is asymptomatic regarding the x-axis
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As x approaches positive infinity, the graph rises without bound.
In situations where the bases are fractions or decimals within 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 greater than 0
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The domain is entirely 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 approaches positive infinity, the line in the 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 vital rules to bear in mind when dealing 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 note it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, deduct the exponents.
For instance, if we need to divide two exponential functions with a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase 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 write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equal to 1.
For example, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are generally utilized to signify exponential growth. As the variable increases, the value of the function increases faster and faster.
Example 1
Let’s examine the example of the growth of bacteria. If we have a culture of bacteria that doubles hourly, then at the end of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x 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 displayed 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 hourly.
Example 2
Also, exponential functions can illustrate exponential decay. If we have a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.
At the end of two hours, we will have one-fourth as much substance (1/2 x 1/2).
After three hours, we will have an eighth as much substance (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 volume of material at time t and t is measured in hours.
As you can see, both of these illustrations follow a comparable pattern, which is the reason they can be depicted using exponential functions.
In 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 whereas the base stays fixed. This means that any exponential growth or decay where the base is different is not an exponential function.
For example, in the matter of compound interest, the interest rate stays the same while the base changes 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 have to enter different values for x and then asses the matching values for y.
Let's check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the values of y increase very rapidly as x increases. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that goes up from left to right and gets steeper as it goes.
Example 2
Chart the following exponential function:
y = 1/2^x
First, let's draw up 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 graph the x-values and y-values on a coordinate plane, it is going to look like the following:
The above is a decay function. As shown, 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 special properties whereby 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 terminology are the powers of an independent variable number. The general form of an exponential series is:
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