Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with multiple values in in contrast to one another. 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 total score. In mathematical terms, the result 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 can be defined as an instrument that takes respective items (the domain) as input and makes particular other pieces (the range) as output. This might be a tool whereby you can obtain multiple snacks for a respective quantity 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 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 put it simply, it is the set 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) could 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).
But, there are particular conditions under which a function cannot be stated. 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 batch of all possible output values for the function. To put it simply, it is the batch of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we might see that the range is all real numbers greater than or equal to 1. No matter what value we apply to x, the output y will always be greater than or equal to 1.
However, just like with the domain, there are particular conditions under which the range may not be specified. For instance, if a function is not continuous at a particular point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range can also be identified via interval notation. Interval notation explains a set of numbers applying two numbers that identify the lower and upper limits. For example, the set of all real numbers in the middle of 0 and 1 might be identified using interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and less than 1 are included in this group.
Similarly, the domain and range of a function can be represented 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 reveals that the function is specified for all real numbers.
The range of this function might be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be classified using graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must find 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 plot these points on a coordinate plane, it will look like this:
As we could watch from the graph, the function is defined for all real numbers. This means that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function creates 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 different types of functions. Let's take a look at 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, every real number might 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 varies among -1 and 1. Further, the function is stated 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 form y= √(ax+b) is stated just for x ≥ -b/a. Consequently, 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 consists of 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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