Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range refer to different values in comparison to each other. For example, let's take a look at the grading system of a school where a student earns an A grade for a cumulative score of 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade shifts with the result. In math, the score 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 might be stated as a machine that catches respective objects (the domain) as input and makes particular other pieces (the range) as output. This can be a tool whereby you might obtain multiple snacks for a specified amount of money.
In this piece, we review the essentials 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 indicate 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, 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 batch of all input values for the function. To clarify, it is the batch 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) might be any real number because we cloud plug in any value for x and acquire a respective output value. This input set of values is required to discover the range of the function f(x).
Nevertheless, there are particular cases under which a function may not be stated. 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 set of all possible output values for the function. To put it simply, it is the set of all y-coordinates or dependent variables. So, applying the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. Regardless of the value we plug in for x, the output y will continue to be greater than or equal to 1.
But, just as with the domain, there are particular terms under which the range must not be stated. 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 via interval notation. Interval notation explains a batch of numbers working with two numbers that represent the bottom and higher limits. For example, the set of all real numbers between 0 and 1 can be identified working with interval notation as follows:
(0,1)
This denotes that all real numbers greater than 0 and lower than 1 are included in this set.
Equally, the domain and range of a function could be classified by applying interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This means that the function is stated for all real numbers.
The range of this function could be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be identified using graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting 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 might watch from the graph, the function is specified for all real numbers. This shows 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 determine the Domain and Range?
The process of finding domain and range values is different for different types of functions. Let's take a look at 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 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 only produces 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 alternates between -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]
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 defined just for x ≥ -b/a. Therefore, 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 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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