Domain and Range Relations

“Domain” are numbers that you give to the function. “Range” means the numbers that the function gives back to you. The first thing one should know about domain and range is that domain is plotted on X-coordinates while Range is plotted on Y-coordinates. Even in cases where you have to find the domain and range from the existing graphs, the rule is the same. You can always think of functions as an existing machine where you put your number and you get a different number out. Some machines can take the numbers that you are giving them and some machines don’t. Some machines will just take out any number from the lot beyond your imagination while some are only known to produce specific numbers. 

Domain and Range Mapping Diagrams

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If there are two existing non-empty sets X and Y, and we have a relation R defined between them as a subset of all the cumulative elements X x Y, the subset is then called as the result of the ‘relation’ existing between the elements of the first set and the elements of the second set. 

Find the Domain and Range of the Relation

In the figure given above, there is a relation from set X to Y. All the rectangular blocks are "related" to the triangular blocks with R A relation may have finite or infinite ordered pairs. If we take a relation from set  X to Y, it is commonly referred to as 'relation on X.' The maximum number of relations that can be defined from set X (having m elements) to Y (having n elements) is equal to 2mn.

Domains are generally easy to find. Finding ranges sometimes can be complicated. In many textbooks, the word "image" is used rather than "range" and "pre-image" for the domain. The reason for that being "range" is used in two different ways in mathematics. It usually means image, the set of values that the function takes on. It is also used for "co-domain."

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Domain and Range Relations Examples

For example, think of graph y=3x+5. You can take any number that you want for x, so the domain for this number will consist of all the real numbers including negative infinity to positive infinity. But, if your function is y=x^2, your domain is still the set of real numbers. But for any real number, whose square results in a positive number. So, the range of the function would be non negative real numbers including zero.

The range is simply the set of all second components of the ordered pairs, with duplicates ignored, so {2; 1; 5; 10}. That eliminates A, B, and D. The domain is the set of all sets that you are allowed to choose from for the first component of the ordered pairs in an itemized list of the relational pairs. In short, if you think of a domain as “all possible inputs” and range as “all possible outputs,” you’ll have the right idea.

Domain and Range of each Relation 

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In all the branches of mathematics, an element in the domain is usually associated with another element of a co-domain. A co-domain, here is a set of all the allowed values while the range is the set of use values for the second component in the ordered pairs. So, here the range turns out to be a subset of the co-domain. 

But for any relation, there are no absolute restrictions on how many elements can exist in a co-domain and one element in the domain can be related to – sometimes they can be zero. Therefore, any element of the domain may or may not be related to any element in the co-domain and similarly, there might not be a relational pair with the value of the first component. In such cases, you wouldn’t be knowing the element is in the domain just with a glance at the relational pairs. 

How to Denote Domain Range Relation 

Domain and range are usually denoted using interval notation, which could look like any of these for both the domain and range, but for this example, let’s just show domain: (smallest value in domain, the largest value in the domain) OR (smallest value in domain, the largest value in the domain) OR (smallest value in domain, the largest value in the domain) OR (smallest number in the domain, largest number in the domain).

FAQ (Frequently Asked Questions)

1. When do we use a Bracket and when do we use a Parenthesis in Relations and Functions – Domain and Range? 

Whether you use a bracket or parenthesis, depends on whether the value it is next to is inclusive in the range or not. For example, a range of (2, infinity) would mean that the smallest y-value approaches, but never actually reaches 2 (since there is a parenthesis) and that the output of the function will continue to increase to infinity. Since infinity is not an actual number, we always use the parenthesis next to infinity (and also, you would put the symbol for infinity, not write it out). These operations are usually very defined, so an expression that’s built from the start is defined to be everywhere. Irrespective of the number that you have in for x, you can compute the value that works with f(x). In this case, the domain for the function are real numbers, from negative infinity to positive infinity. 

2. What are the Main Applications of Domain and Range? 

One of the main applications of mathematics to science, engineering, and business is finding the optimal values for some situations. Everything boils down to find the maximum/minimum of the function. In other words, you just have to find the range of the function where the minimum and maximum values occur. Except if you have the simplest functions, which is not easy to work on. 

These can be found analytically by determining where the derivative of the function is 00 since a differentiable function can only have a maximum or minimum in an open interval when the derivative is 00. The other place where we can use a differentiable function is when it has a minimum or a maximum at the endpoint of the domain.