Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range apply to several values in in contrast to each other. For example, let's check out the grade point calculation of a school where a student gets an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade shifts with the result. In mathematical terms, the total 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 instance, a function might be defined as an instrument that takes specific items (the domain) as input and produces certain other pieces (the range) as output. This might be a instrument whereby you could get multiple treats for a specified amount of money.
In this piece, we discuss the fundamentals of the domain and the range of mathematical functions.
What is the Domain and Range of a Function?
In algebra, the domain and the range refer to the x-values and y-values. For example, 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, 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. In other words, it is the set of all x-coordinates or independent variables. For instance, let's review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can apply any value for x and get a corresponding output value. This input set of values is necessary to figure out the range of the function f(x).
However, there are particular cases under which a function may not be specified. For instance, if a function is not continuous at a particular point, then it is not defined for that point.
The Range of a Function
The range of a function is the batch of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we can see that the range would be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.
But, as well as with the domain, there are specific terms under which the range must not be specified. For instance, 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 might also be represented with interval notation. Interval notation expresses a set of numbers using two numbers that represent the bottom and higher limits. For example, the set of all real numbers in the middle of 0 and 1 can be classified using interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and less than 1 are included in this set.
Equally, the domain and range of a function could be represented via interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) can be represented as follows:
(-∞,∞)
This reveals that the function is defined for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need 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 see 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,∞).
That’s because the function creates 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 is different for different types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is defined for real numbers. Therefore, 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 as y = ax, where a is greater than 0 and not equal to 1. For that reason, every real number could be a possible input value. As the function just produces 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. Also, 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]
Take a 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 just 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 will consistently 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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