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 in contrast to one another. For instance, let's check out grade point averages of a school where a student receives 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 total score. In math, the score is the domain or the input, and the grade is the range or the output.
Domain and range could also be thought of as input and output values. For instance, a function can be stated as an instrument that takes respective items (the domain) as input and produces certain other items (the range) as output. This could be a tool whereby you might buy different snacks for a particular amount of money.
Today, we review 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 cooresponds 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, 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 put it simply, 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) might be any real number because we can apply any value for x and acquire a respective output value. This input set of values is necessary to figure out the range of the function f(x).
But, there are specific cases under which a function cannot be stated. 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 put it simply, it is the batch of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we could see that the range would be all real numbers greater than or the same as 1. Regardless of the value we plug in for x, the output y will always be greater than or equal to 1.
But, as well as with the domain, there are certain terms under which the range may not be defined. For example, if a function is not continuous at a particular point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range can also be classified using interval notation. Interval notation expresses a set of numbers working with two numbers that classify the bottom and higher boundaries. For instance, the set of all real numbers between 0 and 1 can be identified using interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and lower than 1 are included in this group.
Equally, the domain and range of a function could be represented using interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can be represented as follows:
(-∞,∞)
This means that the function is stated for all real numbers.
The range of this function can be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be represented using graphs. So, 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 plot these points on a coordinate plane, it will look like this:
As we could watch from the graph, the function is stated 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 produces 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 is different for multiple types of functions. Let's take a look at some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is stated 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Therefore, each real number might 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 varies 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 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 defined 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 consistently 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 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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