Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range refer to multiple values in comparison to each other. For example, let's consider the grade point calculation of a school where a student gets 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 average grade. Expressed mathematically, the result is the domain or the input, and the grade is the range or the output.
Domain and range can also be thought of as input and output values. For example, a function can be defined as a tool that catches respective pieces (the domain) as input and generates particular other pieces (the range) as output. This might be a machine whereby you can buy several snacks for a specified amount of money.
In this piece, we will teach you the essentials 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. 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, 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 batch of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we might plug in any value for x and get a respective output value. This input set of values is necessary to figure out the range of the function f(x).
Nevertheless, there are specific cases under which a function may not be specified. For instance, if a function is not continuous at a certain 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 put it simply, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.
Nevertheless, as well as with the domain, there are specific terms under which the range may not be specified. For example, if a function is not continuous at a certain point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range could also be classified using interval notation. Interval notation explains a group of numbers working with two numbers that identify the bottom and upper bounds. For instance, the set of all real numbers in the middle of 0 and 1 might be identified working with interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and lower than 1 are included in this batch.
Equally, the domain and range of a function could be classified using interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) can be represented as follows:
(-∞,∞)
This means that the function is defined for all real numbers.
The range of this function might be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be represented with graphs. For instance, let's review the graph of the function y = 2x + 1. Before charting a graph, we must 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 chart these points on a coordinate plane, it will look like this:
As we could look from the graph, the function is stated for all real numbers. This shows us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
That’s because the function generates 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 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. 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 as y = ax, where a is greater than 0 and not equal to 1. Therefore, every real number could be a possible input value. As the function just returns 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 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. For that reason, the domain of the function contains 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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