JEE Main Solved Paper 2017 Question 3
Question: The function $ f:R\to [ -\frac{1}{2},\frac{1}{2} ] $ defined as $ f(x)=\frac{x}{1+x^{2}}, $ is: [JEE Main Solved Paper-2017]
Options:
A) neither injective nor surjective.
B) invertible.
C) injective but not surjective.
D) surjective but not injective
Show Answer
Answer:
Correct Answer: D
Solution:
- $ f:R\to [ -\frac{1}{2},\frac{1}{2} ], $ $ f(x)=\frac{x}{1+x^{2}}\forall x\in R $
$ \Rightarrow $ $ f’(x)=\frac{(1+x^{2}).1-x.2x}{{{(1+x^{2})}^{2}}}=\frac{-(x+1)(x-1)}{{{(1+x^{2})}^{2}}} $
$ \therefore $ From above diagram of $ f(x),f(x) $ is surjective but not injective.
