Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range apply to several values in comparison to each other. For example, let's check out the grading system of a school where a student receives an A grade for a cumulative score of 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the result. In math, the total 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 might be defined as a tool that takes respective items (the domain) as input and makes certain other pieces (the range) as output. This could be a instrument whereby you can buy several treats for a respective quantity of money.
Today, we will teach you 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 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 group of all input values for the function. To clarify, it is the batch 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 cloud plug in any value for x and acquire a corresponding output value. This input set of values is required to find the range of the function f(x).
However, there are specific cases under which a function must 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 set of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we can see that the range is all real numbers greater than or equivalent tp 1. Regardless of the value we assign to x, the output y will continue to be greater than or equal to 1.
But, just as with the domain, there are particular conditions under which the range may not be specified. For example, if a function is not continuous at a particular point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range might also be classified via interval notation. Interval notation expresses a set of numbers working with two numbers that classify the lower and upper bounds. For instance, the set of all real numbers among 0 and 1 can be represented applying interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function could be represented with 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 tells us that the function is stated 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 identified with graphs. For example, let's review the graph of the function y = 2x + 1. Before creating a graph, we have to 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 graph 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 shows us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function produces 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 multiple types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is defined 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, any real number can be a possible input value. As the function just produces positive values, the output of the function contains all positive real numbers.
The domain and range of exponential functions are following:
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Domain = R
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function oscillates among -1 and 1. Further, 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 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 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 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
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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