Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with several values in comparison to one another. For example, let's check out the grading system of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade adjusts with the average grade. In mathematical terms, the score is the domain or the input, and the grade is the range or the output.
Domain and range might also be thought of as input and output values. For example, a function might be defined as a machine that takes particular items (the domain) as input and generates specific other items (the range) as output. This can be a instrument whereby you could obtain several treats for a respective quantity of money.
In this piece, we review the basics 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 check 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, for 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 batch of all x-coordinates or independent variables. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we might plug in any value for x and get a corresponding output value. This input set of values is required to figure out the range of the function f(x).
Nevertheless, there are particular cases under which a function must not be defined. For instance, if a function is not continuous at a particular point, then it is not specified 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 set of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equivalent tp 1. Regardless of the value we assign to x, the output y will continue to be greater than or equal to 1.
However, as well as with the domain, there are certain conditions under which the range may not be defined. For example, if a function is not continuous at a specific point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range might also be represented with interval notation. Interval notation indicates a batch of numbers using two numbers that represent the lower and upper bounds. For instance, the set of all real numbers among 0 and 1 can be identified working with interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function can be classified via 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 stated for all real numbers.
The range of this function might be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified via graphs. For example, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we need to discover 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 could 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 due to the fact that the function produces all real numbers greater than or equal to 1.
How do you figure out the Domain and Range?
The task of finding domain and range values differs for different types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is defined for real numbers. For that reason, the domain for an absolute value function includes 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 might be a possible input value. As the function only produces positive values, the output of the function includes 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. In addition, 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 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 specified 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 Questions 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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