you could write the question directly, instead of enclosing in a document. please do that next time. it is more convenient to see question directly.

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I assume we are talking about relation between x and y : a relation on R --> R

y = f(x) is a function or not.

a) y = f (x ) = -15

This is a many to one function. It is a constant function. Each value of x ∈ R (real numbers) maps to -15. Domain is R and Range is { -15 }.

b) x = 5 relation could be = { ( 5, 1), (5, 0) , (5, 9999), (5, -098) }

Here y can be any value. It means one value of x is mapping onto multiple values of y or F. It is not a function, in the strict sense of a function.

Some times it is referred to as a multi-valued function.

c) Relation = { (1, 0 ) , (1, 4 ) , (-1, 1 ) }

Here x =1 is mapping to 0 and 4. So it is not a function.

d) Relation = { ( 6, 3), (5, -2 ) , (2, 3 ) }

Here each value of x has a unique and one value of y. So it is a function with domain = { 6, 5, 2 } and range = { 3, -2 }

y = f(x) is a function or not.

a) y = f (x ) = -15

This is a many to one function. It is a constant function. Each value of x ∈ R (real numbers) maps to -15. Domain is R and Range is { -15 }.

b) x = 5 relation could be = { ( 5, 1), (5, 0) , (5, 9999), (5, -098) }

Here y can be any value. It means one value of x is mapping onto multiple values of y or F. It is not a function, in the strict sense of a function.

Some times it is referred to as a multi-valued function.

c) Relation = { (1, 0 ) , (1, 4 ) , (-1, 1 ) }

Here x =1 is mapping to 0 and 4. So it is not a function.

d) Relation = { ( 6, 3), (5, -2 ) , (2, 3 ) }

Here each value of x has a unique and one value of y. So it is a function with domain = { 6, 5, 2 } and range = { 3, -2 }