Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range refer to different values in in contrast to each other. For example, let's consider the grade point calculation of a school where a student earns an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade adjusts with the result. In mathematical terms, 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 could be stated as a tool that catches particular items (the domain) as input and makes particular other objects (the range) as output. This might be a machine whereby you might get multiple treats for a respective quantity of money.
In this piece, we review 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 indicate the x-values and y-values. So, let's look at 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, whereas 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 group of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and get a respective output value. This input set of values is necessary to discover the range of the function f(x).
However, there are particular terms under which a function must not be defined. For instance, if a function is not continuous at a specific point, then it is not specified for that point.
The Range of a Function
The range of a function is the group of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.
However, just as with the domain, there are particular conditions under which the range must not be stated. For instance, if a function is not continuous at a specific point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range might also be classified with interval notation. Interval notation indicates a set of numbers using two numbers that identify the lower and upper boundaries. For example, the set of all real numbers in the middle of 0 and 1 could be identified using interval notation as follows:
(0,1)
This means that all real numbers more than 0 and less than 1 are included in this group.
Equally, the domain and range of a function might be classified with interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:
(-∞,∞)
This tells us that the function is specified for all real numbers.
The range of this function could be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be represented using graphs. For instance, let's consider the graph of the function y = 2x + 1. Before charting a graph, we have to determine 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 graph these points on a coordinate plane, it will look like this:
As we might watch from the graph, the function is defined 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 find 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 specified for real numbers. Therefore, 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, every real number can be a possible input value. As the function only delivers positive values, the output of the function consists of 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 shifts among -1 and 1. Further, the function is defined 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 see 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. Therefore, the domain of the function includes all real numbers greater than or equal to b/a. A square function will always result in a non-negative value. So, the range of the function contains 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
Realize 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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