Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range apply to multiple values in in contrast to one another. For instance, let's take a look at 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 total score. Expressed mathematically, 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 instance, a function could be stated as a machine that catches particular pieces (the domain) as input and generates certain other pieces (the range) as output. This can be a tool whereby you can get several treats for a respective quantity of money.
In this piece, we discuss 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 instance, 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, 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. To put it simply, it is the batch of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we might apply any value for x and obtain a respective output value. This input set of values is required to figure out the range of the function f(x).
Nevertheless, there are certain terms under which a function cannot be defined. So, if a function is not continuous at a particular point, then it is not defined for that point.
The Range of a Function
The range of a function is the set of all possible output values for the function. To put it simply, it is the batch of all y-coordinates or dependent variables. For instance, working with the same function y = 2x + 1, we might see that the range will be all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.
Nevertheless, as well as with the domain, there are specific conditions under which the range must not be defined. For instance, if a function is not continuous at a specific point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range might also be identified with interval notation. Interval notation indicates a group of numbers applying two numbers that identify the lower and higher boundaries. For instance, the set of all real numbers among 0 and 1 can be identified working with interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and less than 1 are included in this batch.
Also, the domain and range of a function can be classified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be represented as follows:
(-∞,∞)
This means 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 could also be identified using graphs. So, let's consider the graph of the function y = 2x + 1. Before plotting 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 chart these points on a coordinate plane, it will look like this:
As we might see 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 creates all real numbers greater than or equal to 1.
How do you determine the Domain and Range?
The process of finding domain and range values is different for various 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. 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. Therefore, each real number can 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 alternates among -1 and 1. In addition, 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. Therefore, 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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