The concepts of domain and range emerge from the chapter on relations and functions. Relations and functions are very familiar words for all of us. We know very well that the functions are a subset of the relations. This can also be put up as that relations are the superset of functions. Both the concepts are of extreme importance to us in real-life also. Before understanding what domain and range are, let us understand the concept of functions. A function can be defined as a special type of relationship where there exists only one outcome for each input. Thus, we can say that not all relations are functions, but all functions must necessarily be relations. Let us now discuss the concepts of domain and range.
Real-Life Example of Functions
We can compare the concept of functions with the working of a snack vending machine. When we put a bill of a certain amount inside the machine, we can select our desired snack by paying its price, and then the snack comes out of the vending machine. Likewise, when we talk about functions, we make an input of varieties of numbers for which we get a different set of numbers as the output. Suppose instead of putting a bill of a certain amount, we put a coin inside the vending machine. What will be the result? The vending machine will not give any output. Here, we can compare bills with that of the domain of a function and snacks with that of the range of a function. No matter whatever bill we pay, the output will never turn out to be a cold drink or a soda.
What Do You Mean by a Domain and a Range?
Let a function be defined as f: X→Y, then X is known as the domain of the function f, whereas Y is known as the co-domain of the function f. The set f(X) = {f (y): x belongs to A} is known as the range of the function f. For example, there is relation X = {(4,5), (7,8), (10,2) }, then we have:
Domain = all the numbers of the x-coordinate = {4, 7, 10}
Range = all the numbers of the y-coordinate = {5, 8, 2}
A Minute Difference Between a Co-domain and a Range
Most people get confused with the co-domain and the range. They usually treat both the words as the same concepts and use it interchangeably. However, there is a minute difference between a codomain and a range. Let us understand this difference by taking an example.
Example: Let X = {3, 6, 2, 8} and Y = { 9, 36, 4, 64, 2, 1, 11, 54, 100, 56, 34}
We take the rule f(x) = x.x and then f (3) = 9, f(6) = 36, f(2) = 4 and f(8) = 64
We can see that for each element of X, there exists a unique image in Y.
Here, range of the function f = {3, 6, 2, 8} and codomain of the function f = {9, 36, 4, 64, 2, 1, 11, 54, 100, 56, 34}
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