Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range coorespond with multiple values in comparison to each other. For example, let's check out grade point averages of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade shifts with the total score. In mathematical terms, 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 specified as a tool that catches respective objects (the domain) as input and produces particular other pieces (the range) as output. This might be a machine whereby you might buy different items for a particular amount of money.
Today, we review the fundamentals 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 indicate the x-values and y-values. So, let's check 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 group of all input values for the function. To clarify, it is the set of all x-coordinates or independent variables. For instance, let's review 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 get itsl output value. This input set of values is necessary to discover the range of the function f(x).
But, there are specific conditions under which a function must not be stated. So, 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 batch 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 might see that the range would be all real numbers greater than or the same as 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.
Nevertheless, as well as with the domain, there are specific conditions under which the range must not be stated. For instance, 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 represented via interval notation. Interval notation explains a set of numbers applying two numbers that classify the lower and higher limits. For instance, the set of all real numbers among 0 and 1 might be identified using interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and less than 1 are included in this set.
Similarly, the domain and range of a function could be classified by applying interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) might be classified as follows:
(-∞,∞)
This means that the function is specified for all real numbers.
The range of this function could be classified as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be classified using graphs. For example, let's consider the graph of the function y = 2x + 1. Before creating a graph, we have to 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 graph these points on a coordinate plane, it will look like this:
As we might see from the graph, the function is specified 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 find 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 structure y=|ax+b| is stated for real numbers. Consequently, 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 as y = ax, where a is greater than 0 and not equal to 1. Consequently, every real number could be a possible input value. As the function just delivers 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 shifts between -1 and 1. Also, 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]
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 defined only for x ≥ -b/a. Consequently, 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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