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in Sets, relations and functions by (53.3k points)

The function f : R → [-1/2,1/2] defined as f(x) = x/1 + x2, is :

(A)  injective but not surjective. 

(B)  surjective but not injective. 

(C)  neither injective nor surjective. 

(D)  invertible. 

1 Answer

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by (53.1k points)
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Best answer

 Correct option  (B) surjective but not injective

 The given function is defined as

Therefore, x = 1, -1, which is odd function and there is symmetry about the origin - that is, the function is non-monotonic and noninjective - in the resultant curve as shown in the following figures: 

Any line parallel to x-axis cuts the graph more than one point; hence, the function is many-to-one. Now,

Now, D > 0;1 - 4y2 ≥ 0 . That is, the range is

y ∈ [-1/2,1/2] = codomain

Hence, the function is onto. Therefore, the function is surjective but not injective.

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