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 each other. For instance, 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 result. In mathematical terms, 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 could be defined as a tool that takes specific objects (the domain) as input and makes specific other items (the range) as output. This can be a instrument whereby you might get multiple snacks for a particular amount of money.
Today, we will teach you 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 refer to 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 batch of all input values for the function. To clarify, it is the batch of all x-coordinates or independent variables. For example, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud plug in any value for x and acquire a corresponding output value. This input set of values is needed to find the range of the function f(x).
However, there are particular conditions under which a function must not be defined. So, if a function is not continuous at a certain 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 set of all y-coordinates or dependent variables. So, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or equivalent tp 1. Regardless of the value we apply to x, the output y will always 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 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 can also be classified with interval notation. Interval notation expresses a set of numbers working with two numbers that classify the lower and upper boundaries. For example, the set of all real numbers among 0 and 1 can be identified working with interval notation as follows:
(0,1)
This denotes that all real numbers higher than 0 and less than 1 are included in this set.
Equally, the domain and range of a function could be classified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be classified as follows:
(-∞,∞)
This reveals that the function is defined 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 identified with graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting 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 plot these points on a coordinate plane, it will look like this:
As we can look 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,∞).
That’s 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 differs 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 specified for real numbers. Consequently, 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 can 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 shifts among -1 and 1. Further, 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 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 specified just for x ≥ -b/a. Consequently, the domain of the function contains 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 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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