Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range refer to multiple values in comparison to each other. For example, let's take a look at grade point averages of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the result. In math, the result is the domain or the input, and the grade is the range or the output.
Domain and range could also be thought of as input and output values. For example, a function can be specified as a tool that catches respective items (the domain) as input and produces certain other items (the range) as output. This might be a machine whereby you can buy several treats for a particular amount of money.
Today, we will teach you the fundamentals of the domain and the range of mathematical functions.
What are the Domain and Range of a Function?
In algebra, the domain and the range cooresponds to the x-values and y-values. So, let's view the coordinates for the function f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8).
Here the domain values are all the x coordinates, i.e., 1, 2, 3, and 4, because the range values are all the y coordinates, i.e., 2, 4, 6, and 8.
The Domain of a Function
The domain of a function is a set of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. For instance, let's consider the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we cloud plug in any value for x and obtain a respective output value. This input set of values is required to find the range of the function f(x).
Nevertheless, there are specific conditions under which a function may not be specified. So, if a function is not continuous at a specific point, then it is not stated for that point.
The Range of a Function
The range of a function is the batch of all possible output values for the function. In other words, it is the batch of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we might see that the range is all real numbers greater than or the same as 1. No matter what value we apply to x, the output y will always be greater than or equal to 1.
However, just like with the domain, there are particular conditions under which the range may not be stated. For example, if a function is not continuous at a particular point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range can also be identified via interval notation. Interval notation indicates a batch of numbers working with two numbers that classify the bottom and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 can be represented applying interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and less than 1 are included in this group.
Equally, the domain and range of a function could be classified via interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:
(-∞,∞)
This reveals that the function is defined for all real numbers.
The range of this function might be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be classified via graphs. For example, let's consider the graph of the function y = 2x + 1. Before creating a graph, we have to find all the domain values for the x-axis and range values for the y-axis.
Here are the coordinates: (0, 1), (1, 3), (2, 5), (3, 7). Once we plot these points on a coordinate plane, it will look like this:
As we can look from the graph, the function is specified for all real numbers. This tells us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function produces all real numbers greater than or equal to 1.
How do you figure out the Domain and Range?
The process of finding domain and range values is different for multiple types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is stated for real numbers. Consequently, the domain for an absolute value function contains all real numbers. As the absolute value of a number is non-negative, the range of an absolute value function is y ∈ R | y ≥ 0.
The domain and range for an absolute value function are following:
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Domain: R
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Range: [0, ∞)
For Exponential Functions
An exponential function is written in the form of y = ax, where a is greater than 0 and not equal to 1. For that reason, any real number can be a possible input value. As the function just delivers positive values, the output of the function contains all positive real numbers.
The domain and range of exponential functions are following:
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Domain = R
-
Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function varies between -1 and 1. Further, the function is specified for all real numbers.
The domain and range for sine and cosine trigonometric functions are:
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Domain: R.
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Range: [-1, 1]
Just look at the table below for the domain and range values for all trigonometric functions:
For Square Root Functions
A square root function in the structure y= √(ax+b) is stated only for x ≥ -b/a. Consequently, the domain of the function consists of all real numbers greater than or equal to b/a. A square function always result in a non-negative value. So, the range of the function includes all non-negative real numbers.
The domain and range of square root functions are as follows:
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Domain: [-b/a,∞)
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Range: [0,∞)
Practice Examples on Domain and Range
Find the domain and range for the following functions:
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y = -4x + 3
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y = √(x+4)
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y = |5x|
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y= 2- √(-3x+2)
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y = 48
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