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  • Need explanation for: - Sets, Relations and Functions - JEE Main-2

The function f:\: R\rightarrow \left [ -\frac{1}{2},\frac{1}{2} \right ]   defined  as  f\left ( x \right )= \frac{x}{1+x^{2}},is :

  • Option 1)

     injective but not surjective.

  • Option 2)

    surjective but not injective.

  • Option 3)

     neither injective nor surjective.

  • Option 4)

     invertible.

     

 

Answers (2)

best_answer

As we learnt in

One - One or Injective function -

A line parallel to x - axis cut the curve at most one point.

-

 

 f(x)=\frac{x}{1+x^{2}}        f:R\rightarrow \left[-\frac{1}{2},\frac{1}{2} \right ]

f'(x)=\frac{(1+x^{2})\times1-x\times2x}{(1+x^{2})}=\frac{1+x^{2}-2x^{2}}{(1+x^{2})^{2}}=\frac{1-x^{2}}{(1+x^{2})^{2}}

\therefore    So that \frac{-(x^{2}-1)}{(x^{2}+1)^{2}}

So that it is not strictly increasing or decreasing function.

So that it is not one-one.

Correct option is 2.

 


Option 1)

 injective but not surjective.

This is an incorrect option.

Option 2)

surjective but not injective.

This is the correct option.

Option 3)

 neither injective nor surjective.

This is an incorrect option.

Option 4)

 invertible.

 

This is an incorrect option.

Posted by

divya.saini

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