Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range apply to several values in in contrast to each other. For example, let's check out grade point averages of a school where a student gets 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 result. In mathematical terms, the score 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 stated as an instrument that catches respective pieces (the domain) as input and generates particular other items (the range) as output. This could be a machine whereby you could obtain multiple treats for a respective quantity of money.
Today, we discuss the fundamentals 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 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 group of all input values for the function. In other words, it is the set 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 obtain itsl output value. This input set of values is necessary to figure out the range of the function f(x).
Nevertheless, there are particular conditions under which a function must not be stated. So, if a function is not continuous at a certain point, then it is not defined 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 put it simply, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range will be all real numbers greater than or the same as 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.
However, just like with the domain, there are certain conditions under which the range cannot be stated. For instance, if a function is not continuous at a specific point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range might also be classified via interval notation. Interval notation explains a batch of numbers applying two numbers that represent the bottom and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 can be represented working with interval notation as follows:
(0,1)
This means that all real numbers more than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function might be represented via interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This tells us that the function is defined for all real numbers.
The range of this function can be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be identified via graphs. For example, let's consider the graph of the function y = 2x + 1. Before creating 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 graph these points on a coordinate plane, it will look like this:
As we can look from the graph, the function is defined for all real numbers. This shows us 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 multiple types of functions. Let's consider 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 consists of 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. Consequently, each real number could be a possible input value. As the function only returns 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. 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 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 form y= √(ax+b) is specified 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 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
Find 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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