Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with several values in comparison to one another. For example, let's take a look at 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 changes with the average grade. 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 example, a function could be stated as an instrument that takes specific items (the domain) as input and produces specific other items (the range) as output. This might be a machine whereby you could buy several treats for a respective amount of money.
Today, we review the essentials 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 example, 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, whereas 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. In other words, it is the set of all x-coordinates or independent variables. For instance, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and obtain itsl output value. This input set of values is needed to discover the range of the function f(x).
But, there are particular terms under which a function must not be specified. For example, if a function is not continuous at a specific point, then it is not specified 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 batch of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we can see that the range is all real numbers greater than or equal to 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.
But, just as with the domain, there are particular terms under which the range must not be stated. For example, if a function is not continuous at a certain point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range could also be identified using interval notation. Interval notation expresses a batch of numbers using two numbers that classify the bottom and higher limits. For example, the set of all real numbers among 0 and 1 could be represented applying interval notation as follows:
(0,1)
This denotes that all real numbers greater than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function could be identified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:
(-∞,∞)
This means that the function is specified for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before creating 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 look from the graph, the function is defined 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 differs for various types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is specified 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 as y = ax, where a is greater than 0 and not equal to 1. Therefore, any real number could be a possible input value. As the function only produces 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 oscillates between -1 and 1. Further, 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 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 stated just for x ≥ -b/a. For that reason, the domain of the function contains 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
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 instructor if you need assistance understanding domain and range or the trigonometric subjects. Our Berekley math tutors are practiced educators who focus on work with you when it’s convenient for you and tailor their teaching strategy to suit your learning style. Contact us today at (925) 481-5394 to hear more about how Grade Potential can assist you with achieving your academic goals.
