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 in contrast to each other. For instance, let's consider the grade point calculation of a school where a student receives 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 total score. In math, the result 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 specified as a tool that takes respective items (the domain) as input and generates specific other items (the range) as output. This might be a instrument whereby you could get different snacks for a specified 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 indicate the x-values and y-values. For instance, 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 set of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. For instance, 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 get itsl output value. This input set of values is necessary to discover the range of the function f(x).
However, there are certain terms under which a function may not be defined. For example, 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 set of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range is 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.
Nevertheless, just like with the domain, there are specific terms under which the range may not be defined. For example, if a function is not continuous at a certain point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range could also be represented via interval notation. Interval notation explains a batch of numbers using two numbers that represent the bottom and upper boundaries. For instance, the set of all real numbers in the middle of 0 and 1 could be classified applying interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and lower than 1 are included in this set.
Equally, the domain and range of a function could be represented with interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This means that the function is stated for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be classified with graphs. For example, let's review the graph of the function y = 2x + 1. Before plotting a graph, we have 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 could look from the graph, the function is defined for all real numbers. This tells us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function creates 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 consider some examples:
For Absolute Value Function
An absolute value function in the structure 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, every real number might be a possible input value. As the function only 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 alternates 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 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 contains all real numbers greater than or equal to b/a. A square function will 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 Examples 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
Let Grade Potential Help You Learn Functions
Grade Potential would be happy to connect you with a 1:1 math instructor if you are interested in help understanding domain and range or the trigonometric subjects. Our Anaheim math tutors are experienced professionals who strive to tutor you when it’s convenient for you and personalize their tutoring strategy to fit your learning style. Call us today at (714) 410-2473 to hear more about how Grade Potential can support you with achieving your academic goals.
