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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
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B
injective but not surjective
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C
surjective but not injective
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D
neither injective nor surjective
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Solution

The correct option is B surjective but not injective
The domain of given function is R

When $$x$$ = - 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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