Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
This relation is a function because every element in the domain has one and only one image. Each first element (2, 5, 8, 11, 14, 17) has exactly one corresponding second element. Domain = {2, 5, 8, 11, 14, 17} and Range = {1}.
This relation is a function because every element in the domain has one and only one image. Each first element has a unique corresponding second element. Domain = {2, 4, 6, 8, 10, 12, 14} and Range = {1, 2, 3, 4, 5, 6, 7}.
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This relation is not a function because the element 1 in the domain has two different images (3 and 5). According to the definition, every element must have one and only one image, but here element 1 corresponds to both 3 and 5.
Explanation
As per Definition 5 in the textbook, a relation f from set A to set B is a function if every element of set A has one and only one image in set B. In Example 11(ii), the textbook shows that when the same first element corresponds to two different images, the relation is not a function. Applying this: Parts (i) and (ii) satisfy the definition as each domain element has a unique image. Part (iii) fails because element 1 maps to both 3 and 5.