Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range apply to several values in comparison to one another. For example, let's consider the grading system of a school where a student gets 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 math, the result 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 an instrument that takes particular items (the domain) as input and makes specific other pieces (the range) as output. This could be a machine whereby you might get multiple items for a respective quantity of money.
In this piece, we discuss 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 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, because 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. In other words, it is the batch 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 cloud plug in any value for x and get itsl output value. This input set of values is required to discover the range of the function f(x).
However, there are certain terms under which a function cannot be stated. For instance, 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 group of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range is all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.
However, just like with the domain, there are particular conditions under which the range cannot be stated. For example, if a function is not continuous at a particular point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range could also be classified with interval notation. Interval notation indicates a group of numbers applying two numbers that classify the bottom and higher boundaries. For example, the set of all real numbers between 0 and 1 might be represented using interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and lower than 1 are included in this set.
Similarly, the domain and range of a function can be identified using interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) could be represented as follows:
(-∞,∞)
This reveals that the function is specified for all real numbers.
The range of this function might be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be identified with graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need 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 chart these points on a coordinate plane, it will look like this:
As we could see from the graph, the function is specified for all real numbers. This means that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function creates 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 differs for different 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. Therefore, 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, each real number might be a possible input value. As the function only delivers 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
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function alternates between -1 and 1. Also, 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]
Take a 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 stated 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 contains 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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