Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with different 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 average grade. Expressed mathematically, 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 instance, a function could be stated as a tool that catches specific items (the domain) as input and generates certain other items (the range) as output. This could be a machine whereby you can obtain several treats for a specified quantity of money.
Today, we discuss the fundamentals 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 refer to the x-values and y-values. For instance, 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 group of all input values for the function. To put it simply, it is the group 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 cloud apply any value for x and acquire itsl output value. This input set of values is required to figure out the range of the function f(x).
However, there are specific cases under which a function may not be stated. For example, 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 group of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. No matter what value we apply to x, the output y will continue to be greater than or equal to 1.
Nevertheless, just as with the domain, there are specific terms under which the range may not be defined. For example, if a function is not continuous at a certain 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 indicates a group of numbers using two numbers that identify the lower and higher boundaries. For instance, 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 greater than 0 and lower than 1 are included in this batch.
Also, the domain and range of a function could be represented by applying interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:
(-∞,∞)
This means 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 might also be classified with graphs. For example, let's consider the graph of the function y = 2x + 1. Before creating a graph, we have 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 plot these points on a coordinate plane, it will look like this:
As we might see 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,∞).
That’s because the function produces 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 is different for various types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is specified for real numbers. For that reason, the domain for an absolute value function consists of 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. Consequently, each real number might be a possible input value. As the function only returns 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 alternates 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 stated only 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 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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