Question

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

A
invertible
B
injective but not surjective
C
surjective but not injective
D
neither injective nor surjective

Solution

## The correct option is B surjective but not injectiveThe domain of given function is RWhen $$x$$ = - 1, then $$f($$x$$)$$ =  $$\dfrac{-1}{2}$$When $$x$$= 1,  then$$f($$x$$)$$ =  $$\dfrac{1}{2}$$When $$x$$ $$\in$$ (-1,1)$$f($$x$$)$$  $$\in$$ ($$\dfrac{-1}{2}$$, $$\dfrac{1}{2}$$) When $$x$$ $$\in$$ (-$$\infty$$,  - 1) $$\cup$$ (1,$$\infty$$),  then$$f($$x$$)$$  $$\in$$ ($$\dfrac{-1}{2}$$, $$\dfrac{1}{2}$$)Therefore for all values of domain,  the range of function $$f($$x$$)$$ lies between [$$\dfrac{-1}{2}$$, $$\dfrac{1}{2}$$] => $$f($$x$$)$$  $$\in$$ [$$\dfrac{-1}{2}$$, $$\dfrac{1}{2}$$] Therefore,  the given function is subjective and is not injective Mathematics

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