Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range coorespond with multiple values in in contrast to one another. For example, let's check out grade point averages of a school where a student earns an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the result. In mathematical terms, the score 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 could be defined as a machine that catches respective objects (the domain) as input and generates particular other items (the range) as output. This might be a instrument whereby you could obtain multiple items for a particular amount of money.
Today, we discuss the basics 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. 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 set of all input values for the function. In other words, it is the batch of all x-coordinates or independent variables. So, 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 apply any value for x and get itsl output value. This input set of values is needed to find the range of the function f(x).
However, there are particular terms under which a function cannot be defined. So, if a function is not continuous at a particular point, then it is not stated 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, applying the same function y = 2x + 1, we might see that the range will be all real numbers greater than or the same as 1. Regardless of the value we plug in for x, the output y will continue to be greater than or equal to 1.
Nevertheless, just as with the domain, there are certain conditions under which the range cannot be stated. For instance, if a function is not continuous at a specific point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range could also be represented via interval notation. Interval notation expresses a batch of numbers applying two numbers that represent the bottom and upper boundaries. For example, the set of all real numbers in the middle of 0 and 1 might be represented working with interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and less than 1 are included in this batch.
Similarly, the domain and range of a function might be represented with interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:
(-∞,∞)
This tells us that the function is defined for all real numbers.
The range of this function could be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be represented via 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 can watch from the graph, the function is stated for all real numbers. This tells 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 determine the Domain and Range?
The task of finding domain and range values is different for multiple 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. 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. Therefore, every real number could be a possible input value. As the function only returns 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 shifts between -1 and 1. Also, the function is defined 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 specified 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 consistently 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
Discover 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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