Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range coorespond with several values in comparison to one another. For example, let's consider the grade point calculation 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 adjusts 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 instance, a function can be defined as a tool that takes particular items (the domain) as input and generates certain other pieces (the range) as output. This might be a machine whereby you might get several items for a respective 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 indicate the x-values and y-values. For instance, 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, 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 batch of all input values for the function. To clarify, it is the batch of all x-coordinates or independent variables. For example, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can apply any value for x and acquire a corresponding output value. This input set of values is needed to find the range of the function f(x).
But, there are specific conditions under which a function cannot be specified. For instance, if a function is not continuous at a certain 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 set of all y-coordinates or dependent variables. So, working with the same function y = 2x + 1, we can see that the range is 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.
But, just like with the domain, there are specific terms under which the range may not be stated. For example, if a function is not continuous at a certain point, then it is not stated for that point.
Domain and Range in Intervals
Domain and range can also be identified with interval notation. Interval notation explains a batch of numbers using two numbers that classify the lower and higher boundaries. For instance, the set of all real numbers between 0 and 1 might be classified applying interval notation as follows:
(0,1)
This reveals that all real numbers more than 0 and less than 1 are included in this batch.
Equally, the domain and range of a function can be identified via interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:
(-∞,∞)
This tells us that the function is stated for all real numbers.
The range of this function could be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be represented via graphs. So, let's review the graph of the function y = 2x + 1. Before charting a graph, we must discover 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 might 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 creates all real numbers greater than or equal to 1.
How do you find the Domain and Range?
The process of finding domain and range values is different for various types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is stated for real numbers. Therefore, 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. For that reason, each real number might be a possible input value. As the function just produces 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
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Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function shifts between -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 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 specified only for x ≥ -b/a. Therefore, 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 Questions 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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