Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range coorespond with several values in comparison to each other. For example, let's take a look at grade point averages of a school where a student earns 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 adjusts with the average grade. Expressed mathematically, 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 could be stated as a tool that catches particular objects (the domain) as input and makes specific other pieces (the range) as output. This might be a machine whereby you might buy multiple items for a specified amount of money.
Today, we review 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 cooresponds to the x-values and y-values. For example, 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, for 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 put it simply, 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) might be any real number because we cloud apply any value for x and get a respective output value. This input set of values is necessary to find the range of the function f(x).
Nevertheless, there are particular cases under which a function cannot be specified. For example, if a function is not continuous at a specific point, then it is not stated 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 be specific, it is the group of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.
Nevertheless, just like with the domain, there are certain conditions under which the range may not be defined. For example, 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 identified with interval notation. Interval notation expresses a batch of numbers working with two numbers that identify the lower and higher bounds. For example, the set of all real numbers among 0 and 1 can be identified working with interval notation as follows:
(0,1)
This reveals that all real numbers higher than 0 and lower than 1 are included in this batch.
Also, the domain and range of a function could be classified by applying interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) can be represented as follows:
(-∞,∞)
This reveals that the function is stated for all real numbers.
The range of this function might be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified using graphs. For instance, let's review the graph of the function y = 2x + 1. Before plotting a graph, we need 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 can 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 due to the fact that the function generates 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 various types of functions. Let's watch 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 contains 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. For that reason, any real number can be a possible input value. As the function just returns 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 varies 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 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. Therefore, 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 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 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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