Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range apply to several values in in contrast to one another. For instance, let's check out 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 a cumulative score of 81 - 90, and so on. Here, the grade changes with the total score. In mathematical terms, the total is the domain or the input, and the grade is the range or the output.
Domain and range might also be thought of as input and output values. For instance, a function could be defined as a machine that catches specific objects (the domain) as input and produces certain other objects (the range) as output. This can be a machine whereby you can get multiple items for a respective amount of money.
Today, 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 cooresponds to 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 clarify, it is the group of all x-coordinates or independent variables. So, 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 acquire a corresponding output value. This input set of values is required to find the range of the function f(x).
But, there are particular conditions under which a function may not be defined. 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. In other words, it is the batch of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. Regardless of the value we apply to x, the output y will continue to be greater than or equal to 1.
But, just as with the domain, there are specific terms under which the range must not be defined. For instance, 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 represented via interval notation. Interval notation indicates a set of numbers working with two numbers that classify the bottom and upper bounds. For example, the set of all real numbers in the middle of 0 and 1 could be classified applying interval notation as follows:
(0,1)
This means that all real numbers greater than 0 and less than 1 are included in this set.
Also, the domain and range of a function might be represented by applying interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) might be represented as follows:
(-∞,∞)
This means that the function is defined for all real numbers.
The range of this function can be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be identified using graphs. So, 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 see 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 generates all real numbers greater than or equal to 1.
How do you figure out the Domain and Range?
The task 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 structure y=|ax+b| is defined 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. Consequently, each real number can be a possible input value. As the function just produces 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 shifts between -1 and 1. In addition, the function is defined 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 structure y= √(ax+b) is specified just 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 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
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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