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1.3. DOMAIN AND RANGE
Defining domain and range of relation A relation R between the elements of a set X and the elements of a set Y is the set of pairs (x, y) where x is an element of X and y is an element of Y. The relations nay not include all pairs giving us a correspondence between some values of x and some values of y only. There are always two sets associated with a relation R:
(1) the set of values of the variable x which have a pair in the relation R; (2) the set of values of the variable y which have a pair in the relation R. Below we give more precise definition.
1.3.1. DEFINITION. Let R be a relation. Then R is a subset of the set of all pairs
{(x, y)|x belongs X and y belongs to Y }.
The domain of R is the set
{x|x belongs to X and there exists y in Y such that x is related to y}.
The range of R is the set
{y|y belongs to Y and there exists x in X which is related to y}.
1.3.2. EXAMPLE.
In the above figure the oval-shaped region represents a relation and we can see that the number 5 belongs to the domain of the relation because the vertical line passing through 5 in the x-axis intersects the region. The same is true for each number between 1 and 7 including 1 and 7. So the domain is the closed interval [1, 7].
Finding domains and ranges of relations
1.3.4. EXERCISES.
- Exercise. Find the domain and the range of the relation
R = {(2, 5), (4, 3), (6, 1), (2, 7)}.
Go to answer 1
- Exercise. Find the domain and the range of the relation by the equation 2 x + 3y = 5. Go to answer 2
- Exercise. Find the domain and the range of the relation by the equation xy = 1.
Go to answer 3
- Exercise. Find the domain and the range of the relation by the equation y = x^2 โ 3.
Go to answer 4
- Exercise. Find the domain and the range of the relation by the equation y = (^) xxโ 2.
Go to answer 5
- Exercise. Find the domain and the range of the relation by the equation y^2 = x โ 3.
Go to answer 6
1.3.7. ANSWERS.
- Answer to Exercise 1. The domain of R is 2, 4 , 6 because the numbers 2, 4, 6 appear as the first elements of the pairs in R. The range of R is { 5 , 3 , 1 , 7 } because the numbers 5, 3, 1, 7 appear as the second elements of the pairs in R. Go back 1
- Answer to Exercise 2. The domain of R is the set of all real numbers. If x is a real number then solving the equation for y we see that x is related to y = 53 โ 23 x. For instance x = 2 is related to y = 13. The range of R is the set of all real numbers. If y is a real number then solving the equation for x we obtain that x = 52 โ 32 y is related to y. For instance if y = 3 then x = โ2 is related to y = 3. Go back 2
- Answer to Exercise 3. The domain and the range of R is the set of all real numbers except for the number 0. We explain how to find the domain only. If x = 0 then for every value of y we have 0y = 0. It means that there is no value of y such that0y = 1. Thus the number 0 does not belong to the domain. If x 6 = 0 then x is related to y = (^) x^1. Go back 3
- Answer to Exercise 4. The domain of R is the set of all real numbers because for every value of x the number x is related to y = x^2 โ 3. The range of R is the interval [โ 3 , x). If x^2 โฅ 0 then x^2 โ 3 โฅ โ3 and y โฅ โ3. So we see that if y < โ3 then there is no x such that y = x^2 โ 3. It means that y does not belong to the range. If y โฅ โ3 then y + 3 โฅ 0 and the square root of y + 3 is defined. So x equal to
y + 3 is related to y. Go back 4
- Answer to Exercise 5. The domain is the set of all real numbers but, the number 2 because substitution x = 2 leads to dividing by 0. The range is the set of all real numbers but the number 1 because after solving the equation for x we obtain x = (^) (y^2 โy1) which is undefined for y = 1. Go back 5
- Answer to Exercise 6. The domain is the interval [3, X)and the range is the set of all real numbers. Since for every value of y we have y^2 โฅ 0 the value of x needs to satisfy the inequality x โ 3 โฅ 0 which gives x โฅ 3.
Go back 6
