5. Classify each of the following as a function, a partial function, or not a fu

Question

5. Classify each of the following as a function, a partial function, or not a function. (If it's both a function and a partial function, choose "function"-the more exact classification.) a) f (1, a)) for domain space 1 and range space (a, b}. b) f(x) - 1/x when both domain and range space are the Real numbers (R) c) f(x)sin(x) when domain space is the Natural numbers (N, the non-negative integers) and range space is R. d) f(x)x% when both domain and range space are R. (Recall that the square root of a positive number n is n.) e) f(x, y)- max(x, y) for domain space R x R and range space R. f) f = {(, a), (, b), (, b)) for domain space 1, 2) x {1, 2) and range space fa, b. for domain space 1) x 1,2) and range space (a, b)

Explanation / Answer

Function: A function f: X-> Y is defined as relation between X and Y such that for each element in X there exists a unique element in Y. X is domain and Y is range.

Partial Function : A partial function is a function f: X'->Y for some subset X' of X. It leaves the restriction that each element of X should map to some element of Y.

(a) f = {(1,a)} for domain space {1} and range space {a, b}

Here X = {1}     Y = {a,b}

As element of X matches to unique element in Y and every element of X is covered therefore it is a function. It is one -one but but onto as every element of Y has not been mapped.

(b) f(x) = 1/x when both domain and range are real numbers

When x = 0 , f(x) becomes undefined or infinity i.e. our function f(x) is not defined on x=0 and hence it is a partial function as our function is not defined for each element in domain.

(c) f(x) = sin(x) when domain is natural numbers(N) and range is R

It is a function as sinx is a continuous function and is defined on every natural number as well. However, it is not one-one as for x = 180 and x=360 sinx maps to same value in range i.e. 0.

It is neither onto as for function to be onto every element of Y has to be covered in mapping. But sinx has range [-1 1] . Element 2 in range has no pre-image . Therefore, it is not onto function.

(d) f(x) = x1/2 when both domain and range space are R.

When x = 9 function will map to both -3 and 3 in Y. Hence, element in X doesnot map to unique element in Y. Therefore, it is not a function from above given definition.

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