Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range refer 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 an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade adjusts with the result. 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 might be stated as a machine that takes respective objects (the domain) as input and generates certain other pieces (the range) as output. This can be a instrument whereby you might buy different snacks for a respective quantity of money.
In this piece, we will teach you 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 set of all input values for the function. In other words, it is the group of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we might plug in any value for x and get a respective output value. This input set of values is needed to figure out the range of the function f(x).
Nevertheless, there are particular cases under which a function cannot be defined. So, if a function is not continuous at a particular point, then it is not stated for that point.
The Range of a Function
The range of a function is the batch of all possible output values for the function. To be specific, it is the batch 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 assign to x, the output y will always be greater than or equal to 1.
But, just like with the domain, there are certain conditions under which the range may not be specified. 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 might also be represented via interval notation. Interval notation expresses a group of numbers applying two numbers that represent the lower and upper limits. For example, the set of all real numbers among 0 and 1 could be represented working with interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and lower than 1 are included in this group.
Also, the domain and range of a function might be identified using interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be represented as follows:
(-∞,∞)
This means 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 could also be represented with graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must 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 plot 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 shows us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function generates all real numbers greater than or equal to 1.
How do you find the Domain and Range?
The task of finding domain and range values is different for various 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 includes 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. Therefore, every real number could be a possible input value. As the function just delivers 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
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function alternates between -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 defined just 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 will consistently 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 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
Let Grade Potential Help You Master Functions
Grade Potential would be happy to match you with a private math teacher if you are interested in help understanding domain and range or the trigonometric subjects. Our Cincinnatti math tutors are skilled professionals who strive to tutor you on your schedule and customize their teaching strategy to match your learning style. Call us today at (513) 654-3123 to hear more about how Grade Potential can support you with obtaining your learning goals.
