Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range apply to multiple values in in contrast to each other. For instance, let's consider the grading system 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 total score. In mathematical terms, the score 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 can be specified as a machine that takes specific items (the domain) as input and makes certain other pieces (the range) as output. This can be a tool whereby you might buy multiple snacks for a specified amount 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. 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, 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 set of all input values for the function. To clarify, it is the group 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) can be any real number because we can apply any value for x and get itsl 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 cannot be defined. For instance, if a function is not continuous at a certain 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, working with the same function y = 2x + 1, we can see that the range would be all real numbers greater than or equivalent tp 1. No matter what value we apply to x, the output y will always be greater than or equal to 1.
Nevertheless, as well as with the domain, there are specific terms under which the range must 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 could also be represented via interval notation. Interval notation explains a group of numbers using two numbers that classify the lower and upper bounds. For instance, the set of all real numbers among 0 and 1 could be identified applying interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and less than 1 are included in this set.
Equally, the domain and range of a function could be represented using interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:
(-∞,∞)
This means that the function is specified for all real numbers.
The range of this function might be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified via graphs. So, let's review the graph of the function y = 2x + 1. Before plotting a graph, we need to determine 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 could see from the graph, the function is stated for all real numbers. This tells 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 figure out the Domain and Range?
The process of finding domain and range values is different for multiple types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is specified for real numbers. Therefore, the domain for an absolute value function consists of 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. For that reason, any real number might be a possible input value. As the function just delivers 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
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function oscillates between -1 and 1. Also, 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]
Take a 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 stated only for x ≥ -b/a. For that reason, 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 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
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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