Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range coorespond with different values in comparison to each other. For instance, let's take a look at grade point averages of a school where a student receives an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the average grade. In math, the total 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 instance, a function might be defined as a tool that takes specific items (the domain) as input and makes specific other items (the range) as output. This can be a tool whereby you can obtain several treats for a particular amount of money.
Today, we review 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 cooresponds to the x-values and y-values. For instance, 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, 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 batch of all input values for the function. In other words, it is the set of all x-coordinates or independent variables. So, let's consider 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 obtain a corresponding output value. This input set of values is necessary to discover the range of the function f(x).
Nevertheless, there are specific terms under which a function must not be stated. For example, if a function is not continuous at a specific 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 be specific, it is the set of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we can see that the range will be all real numbers greater than or equal to 1. Regardless of the 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 certain conditions under which the range must not be specified. For instance, if a function is not continuous at a particular point, then it is not defined for that point.
Domain and Range in Intervals
Domain and range could also be classified using interval notation. Interval notation explains a set of numbers applying two numbers that classify the lower and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 could be classified working with interval notation as follows:
(0,1)
This denotes that all real numbers greater than 0 and lower than 1 are included in this batch.
Similarly, the domain and range of a function can be identified 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 means that the function is specified 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 represented with graphs. For example, let's review 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 chart these points on a coordinate plane, it will look like this:
As we could watch 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 due to the fact that the function generates 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 is different 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 defined 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. Consequently, any real number could be a possible input value. As the function only returns 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 alternates between -1 and 1. In addition, 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 structure y= √(ax+b) is stated only for x ≥ -b/a. Consequently, the domain of the function consists of 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 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
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
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