Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
To put it simply, domain and range refer to several values in in contrast to one another. For example, let's check out 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 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 could be specified as a tool that catches respective pieces (the domain) as input and generates specific other objects (the range) as output. This can be a tool whereby you could buy several snacks for a respective quantity of money.
Today, we review 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 instance, 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, 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 put it simply, it is the batch of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can plug in any value for x and acquire itsl output value. This input set of values is necessary to discover the range of the function f(x).
But, there are particular cases under which a function must not be specified. For example, if a function is not continuous at a certain point, then it is not stated for that point.
The Range of a Function
The range of a function is the group of all possible output values for the function. To be specific, it is the batch 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 equal to 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.
However, just as with the domain, there are certain conditions under which the range cannot be stated. 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 can also be identified using interval notation. Interval notation explains a set of numbers using two numbers that classify the lower and upper bounds. For instance, the set of all real numbers between 0 and 1 can be identified using interval notation as follows:
(0,1)
This means 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 with interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can be identified as follows:
(-∞,∞)
This means that the function is stated for all real numbers.
The range of this function could be represented as follows:
(1,∞)
Domain and Range Graphs
Domain and range might also be classified via graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we need 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 plot 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 tells us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is because the function produces 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 different types of functions. Let's take a look at 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, any real number might be a possible input value. As the function just returns 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 oscillates between -1 and 1. Also, 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 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 just for x ≥ -b/a. For that reason, the domain of the function contains 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 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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