The domain the a role f ( x ) is the set of all worths for which the duty is defined, and also the selection of the function is the set of all values that f takes.

(In grammar school, you probably dubbed the domain the replacement set and the range the solution set. They may likewise have been called the input and output that the function.)

example 1:

think about the function shown in the diagram.


Here, the domain is the set A , B , C , E . D is no in the domain, due to the fact that the duty is not characterized for D .

The range is the collection 1 , 3 , 4 . 2 is not in the range, since there is no letter in the domain the gets mapped come 2 .

girlfriend can likewise talk around the domain the a relationship , wherein one aspect in the domain may acquire mapped to much more than one aspect in the range.

instance 2:

think about the relation ( 0 , 7 ) , ( 0 , 8 ) , ( 1 , 7 ) , ( 1 , 8 ) , ( 1 , 9 ) , ( 2 , 10 ) .

Here, the relationship is given as a collection of notified pairs. The domain is the collection of x -coordinates, 0 , 1 , 2 , and also the selection is the set of y -coordinates, 7 , 8 , 9 , 10 . Note that the domain aspects 1 and 2 are connected with more than one variety elements, for this reason this is no a function.

But, much more commonly, and especially when taking care of graphs ~ above the coordinate plane, we are came to with functions, where each aspect of the domain is linked with one aspect of the range. (See The Vertical heat Test .)

example 3:

The domain the the role

f ( x ) = 1 x

is all real numbers other than zero (since at x = 0 , the duty is undefined: department by zero is not allowed!).

The range is additionally all real numbers except zero. You have the right to see the there is some allude on the curve for every y -value other than y = 0 .


domain names can additionally be explicitly specified, if there room values for which the function could be defined, yet which we don"t want to take into consideration for some reason.

instance 4:

The adhering to notation reflects that the domain of the function is limited to the interval ( − 1 , 1 ) .

f ( x ) = x 2 ,         − 1 x 1

The graph the this duty is together shown. Keep in mind the open circles, which display that the function is not defined at x = − 1 and x = 1 . The y -values range from 0 approximately 1 (including 0 , however not consisting of 1 ). For this reason the selection of the role is

0 ≤ y 1 .


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