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 comparison 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 changes 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 example, a function can be specified as an instrument that catches particular pieces (the domain) as input and generates specific other pieces (the range) as output. This can be a instrument whereby you might buy several treats for a respective quantity of money.
Today, we discuss the basics 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. 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 set of all input values for the function. To put it simply, it is the group of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we might plug in any value for x and obtain a corresponding output value. This input set of values is necessary to find the range of the function f(x).
But, there are specific terms under which a function may 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 be specific, it is the group of all y-coordinates or dependent variables. For example, using the same function y = 2x + 1, we can see that the range would be all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.
But, just like with the domain, there are certain conditions under which the range may not be stated. For example, 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 identified using interval notation. Interval notation expresses a set of numbers working with two numbers that identify the lower and higher boundaries. For instance, the set of all real numbers in the middle of 0 and 1 might be classified applying interval notation as follows:
(0,1)
This reveals that all real numbers greater than 0 and lower than 1 are included in this group.
Similarly, the domain and range of a function could be identified by applying interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) can be represented as follows:
(-∞,∞)
This means that the function is defined for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range can also be classified with graphs. For instance, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need to find 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 look 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 various types of functions. Let's watch some examples:
For Absolute Value Function
An absolute value function in the structure y=|ax+b| is stated 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. Therefore, every real number might be a possible input value. As the function only produces positive values, the output of the function includes 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 varies between -1 and 1. In addition, 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 stated just 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 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
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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