Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range apply to multiple values in comparison to one another. For example, let's consider grade point averages of a school where a student earns 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 changes with the average grade. 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 example, a function might be defined as a tool that catches specific pieces (the domain) as input and makes certain other pieces (the range) as output. This could be a instrument whereby you could get multiple snacks for a respective 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 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, 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. In other words, it is the batch 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) might be any real number because we might plug in any value for x and get a corresponding output value. This input set of values is necessary to find the range of the function f(x).
Nevertheless, there are certain cases under which a function cannot be defined. So, 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 group of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. For instance, working with the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.
However, just as with the domain, there are particular 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 specified for that point.
Domain and Range in Intervals
Domain and range could also be classified using interval notation. Interval notation explains a group of numbers applying two numbers that classify the bottom and upper boundaries. For example, the set of all real numbers in the middle of 0 and 1 might be represented using interval notation as follows:
(0,1)
This denotes that all real numbers higher than 0 and lower than 1 are included in this batch.
Equally, the domain and range of a function can be identified via interval notation. So, let's review 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 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 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 chart these points on a coordinate plane, it will look like this:
As we could look from the graph, the function is specified 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 figure out the Domain and Range?
The process 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 stated for real numbers. For that reason, 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 in the form of 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
-
Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function varies among -1 and 1. Further, 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 see 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 defined only for x ≥ -b/a. Consequently, the domain of the function consists of 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 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 Questions 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
Let Grade Potential Help You Master Functions
Grade Potential would be happy to pair you with a private math teacher if you need help understanding domain and range or the trigonometric concepts. Our Kent math tutors are skilled educators who strive to partner with you when it’s convenient for you and tailor their tutoring strategy to match your needs. Reach out to us today at (330) 590-5311 to learn more about how Grade Potential can help you with achieving your academic goals.
