Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In simple terms, domain and range coorespond with different values in in contrast to each other. For instance, let's take a look at the grading system 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 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 could also be thought of as input and output values. For example, a function could be specified as a machine that catches specific items (the domain) as input and produces specific other pieces (the range) as output. This could be a machine whereby you could buy multiple items for a specified amount of money.
Today, we discuss the fundamentals 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. So, 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, 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 set of all input values for the function. To put it simply, it is the set of all x-coordinates or independent variables. For example, let's review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we might plug in any value for x and acquire a corresponding output value. This input set of values is needed to figure out the range of the function f(x).
However, there are specific conditions under which a function cannot be stated. For instance, if a function is not continuous at a particular point, then it is not stated 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 group of all y-coordinates or dependent variables. So, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. Regardless of the value we assign to x, the output y will continue to be greater than or equal to 1.
Nevertheless, just like with the domain, there are specific conditions under which the range cannot be specified. For example, 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 identified with interval notation. Interval notation explains a batch of numbers working with two numbers that classify the bottom and upper boundaries. For example, the set of all real numbers in the middle of 0 and 1 might be identified applying interval notation as follows:
(0,1)
This means that all real numbers higher than 0 and lower than 1 are included in this batch.
Also, the domain and range of a function might be represented using interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) might be represented as follows:
(-∞,∞)
This reveals that the function is specified for all real numbers.
The range of this function could be identified as follows:
(1,∞)
Domain and Range Graphs
Domain and range could also be identified using graphs. So, let's consider the graph of the function y = 2x + 1. Before charting a graph, we have 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 plot these points on a coordinate plane, it will look like this:
As we might watch from the graph, the function is specified for all real numbers. This shows us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
That’s because the function produces 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 specified 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, any real number could be a possible input value. As the function just delivers 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
-
Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function oscillates 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]
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 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 always result in a non-negative value. So, the range of the function consists of 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
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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