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 comparison to one another. For example, let's check out grade point averages of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the average grade. 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 can be stated as a machine that catches specific pieces (the domain) as input and makes specific other pieces (the range) as output. This might be a instrument whereby you could get several snacks for a respective quantity of money.
In this piece, we review the basics 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. 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 set of all input values for the function. To put it simply, it is the group of all x-coordinates or independent variables. So, 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 required to find 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 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. In other words, it is the batch of all y-coordinates or dependent variables. So, using the same function y = 2x + 1, we can see that the range is all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.
However, as well as with the domain, there are specific terms under which the range must not be defined. For example, if a function is not continuous at a specific point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range could also be identified using interval notation. Interval notation expresses a set of numbers using two numbers that identify the lower and higher limits. For instance, the set of all real numbers between 0 and 1 might be represented working with interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and lower than 1 are included in this batch.
Similarly, the domain and range of a function might be classified using interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:
(-∞,∞)
This reveals that the function is specified for all real numbers.
The range of this function might be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be identified via graphs. For example, let's consider the graph of the function y = 2x + 1. Before plotting 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 graph these points on a coordinate plane, it will look like this:
As we could see from the graph, the function is stated for all real numbers. This means 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 is different for various types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is specified 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 as y = ax, where a is greater than 0 and not equal to 1. For that reason, any 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 shifts between -1 and 1. In addition, 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]
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 form y= √(ax+b) is defined just for x ≥ -b/a. For that reason, 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 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
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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