Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with multiple values in in contrast to one another. For instance, 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 result. Expressed mathematically, the total 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 catches particular items (the domain) as input and generates specific other objects (the range) as output. This can be a machine whereby you can obtain different items for a specified quantity of money.
Today, we discuss the basics 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. So, let's look at 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 group of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we might apply any value for x and acquire a corresponding output value. This input set of values is required to figure out the range of the function f(x).
However, there are particular terms under which a function must not be defined. So, 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 set of all possible output values for the function. To put it simply, it is the batch of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we can see that the range would be all real numbers greater than or equivalent tp 1. No matter what value we apply to x, the output y will continue to be greater than or equal to 1.
Nevertheless, just like with the domain, there are particular conditions under which the range may not be specified. For example, if a function is not continuous at a particular point, then it is not specified for that point.
Domain and Range in Intervals
Domain and range can also be identified with interval notation. Interval notation expresses a set of numbers working with two numbers that represent the bottom and higher limits. For instance, the set of all real numbers between 0 and 1 can be represented 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 set.
Equally, the domain and range of a function might be identified by applying 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 reveals 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 with graphs. For instance, let's review the graph of the function y = 2x + 1. Before charting 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 plot these points on a coordinate plane, it will look like this:
As we could watch from the graph, the function is stated for all real numbers. This shows 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 figure out the Domain and Range?
The process 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 form y=|ax+b| is defined for real numbers. Consequently, 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, each real number could be a possible input value. As the function only delivers positive values, the output of the function contains 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]
Take a 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 specified only 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 will always result in a non-negative value. So, the range of the function includes 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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