Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range apply to different values in in contrast to each other. For example, let's take a look at the grade point calculation of a school where a student gets 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 shifts with the result. Expressed mathematically, the total 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 example, a function might be stated as an instrument that catches respective items (the domain) as input and produces certain other objects (the range) as output. This can be a machine whereby you can get multiple treats for a particular quantity of money.
Today, we review the basics 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. For instance, let's view 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 clarify, 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 apply any value for x and acquire a corresponding output value. This input set of values is needed to discover the range of the function f(x).
But, there are certain terms under which a function may not be defined. For instance, if a function is not continuous at a particular 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 group of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we could see that the range will be all real numbers greater than or equal to 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.
However, as well as 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 particular point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range can also be classified with interval notation. Interval notation indicates a batch of numbers applying two numbers that represent the bottom and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 could be identified using interval notation as follows:
(0,1)
This denotes that all real numbers greater than 0 and less than 1 are included in this batch.
Equally, the domain and range of a function could be identified by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:
(-∞,∞)
This means that the function is specified 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 classified with graphs. For example, let's review the graph of the function y = 2x + 1. Before plotting a graph, we have 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 can look from the graph, the function is stated for all real numbers. This means that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
That’s 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 is different for different types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is defined for real numbers. Consequently, 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. Consequently, every real number can be a possible input value. As the function only returns 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
-
Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function varies between -1 and 1. Also, the function is stated 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 stated only for x ≥ -b/a. Consequently, the domain of the function includes 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 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
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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