Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range apply to different values in in contrast to one another. For example, let's check out grade point averages 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 shifts with the average grade. Expressed mathematically, 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 can be stated as a machine that catches respective objects (the domain) as input and produces certain other pieces (the range) as output. This might be a machine whereby you could get multiple snacks for a respective amount of money.
Today, we will teach you the essentials 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 refer to the x-values and y-values. So, 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 batch of all input values for the function. To put it simply, it is the group of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we cloud plug in any value for x and get a respective output value. This input set of values is necessary to discover the range of the function f(x).
However, there are specific cases under which a function must not be stated. So, if a function is not continuous at a particular point, then it is not defined 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 put it simply, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. Regardless of the value we plug in for x, the output y will always be greater than or equal to 1.
However, just as with the domain, there are certain conditions under which the range cannot be defined. For example, if a function is not continuous at a certain point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range could also be classified via interval notation. Interval notation indicates a group of numbers working with two numbers that classify the bottom and upper boundaries. For example, the set of all real numbers among 0 and 1 could be classified working with interval notation as follows:
(0,1)
This means 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 classified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This reveals that the function is defined for all real numbers.
The range of this function might be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be identified via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before plotting a graph, we have to 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 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,∞).
This is because the function creates all real numbers greater than or equal to 1.
How do you figure out the Domain and Range?
The task of finding domain and range values differs for various types of functions. Let's take a look at some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is defined for real numbers. Therefore, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. For that reason, every real number might be a possible input value. As the function just returns 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 among -1 and 1. Also, 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 look at 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 stated only 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 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
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