Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range refer to different values in comparison to one another. For example, let's consider grade point averages of a school where a student receives 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 mathematical terms, the result is the domain or the input, and the grade is the range or the output.
Domain and range can also be thought of as input and output values. For example, a function could be stated as an instrument that catches specific objects (the domain) as input and makes certain other items (the range) as output. This could be a tool whereby you could obtain multiple items for a specified amount of money.
Today, we will teach you the fundamentals of the domain and the range of mathematical functions.
What are the Domain and Range of a Function?
In algebra, the domain and the range indicate the x-values and y-values. For example, 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, 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 group of all input values for the function. To clarify, it is the batch of all x-coordinates or independent variables. For example, let's review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we can apply any value for x and get a respective output value. This input set of values is required to find the range of the function f(x).
However, there are certain cases under which a function may not be defined. For instance, if a function is not continuous at a particular point, then it is not specified for that point.
The Range of a Function
The range of a function is the batch of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.
However, as well as with the domain, there are specific terms under which the range must not be stated. For example, if a function is not continuous at a specific point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range might also be classified with interval notation. Interval notation indicates a group of numbers using two numbers that represent the bottom and higher bounds. For example, the set of all real numbers in the middle of 0 and 1 can be represented applying interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and less than 1 are included in this group.
Equally, the domain and range of a function might be represented by applying interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:
(-∞,∞)
This reveals that the function is stated for all real numbers.
The range of this function might be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be classified via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting 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 graph these points on a coordinate plane, it will look like this:
As we could look from the graph, the function is stated 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 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 differs for different types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is stated for real numbers. Consequently, the domain for an absolute value function contains 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. Consequently, any real number could 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
-
Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function shifts 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 only for x ≥ -b/a. Therefore, 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
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
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