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Let N denote the set of all non-negative integers and Z denote the set of all integers. The function f : Z → N given by f(x) = |x| is:

Let N denote the set of all non-negative integers and Z denote the set of all integers. The function f : Z → N given by f(x) = |x| is: Correct Answer Onto but not one-one

Concept:

One–One Function or Injective Function:
A function f: A → B is said to be a one–one function, if different elements in A have different images or associated with different elements in B i.e if

f (x1) = f (x2) ⇒  x1 = x2, ∀ x1, x2 ∈ A.

Into Function:

Any function f: A → B is said to be into function if there exist at least one element in B which does not has a pre-image in A, then the function f is said to be into function.

If Range of function f ⊂ Co-domain of function f, then f is into

Many-one Function:

Any function f: A → B is said to be many-one, if two (or more than two) distinct elements in A have same images in B.

Onto Function / Surjective Function :

Any function f: A → B is said to be onto if every element in B has atleast one pre-image in A.

If Range of function f = Co-domain of function f, then f is onto

 

Calculation:

Here, f(x) = |x|

f(x) = -x,  when x < 0 and f(x) = x, when x > 0

 

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It cuts the line parallel to the x-axis at two points so, it's not one-one function.

Given function is linear and it is defined Z → N

Hence the range of the function is a natural number.

Therefore co-domain = range

So, the given function is onto.

Hence, option (2) is correct.

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