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The function f:R to [-(1)/(2),(1)/(2)] d...

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

A

invertible

B

injective but not surjective

C

surjective but not injective

D

neither injective nor surjective

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `f(x)=(x)/(1+x^(2))`
`therefore f((1)/(x))=(x)/(1+(1)/(x^(2)))=(x)/(1+x^(2))=f(x)`
` therefore f((1)/(2))=f(2) or f((1)/(3))=f(3) ` and so on.
So, f(x) is many-one function.
Again, let `y=f(x)rArr y=(x)/(1+x^(2))`
`rArr y+x^(2)y=x rArr yx^(2)-x+y=0`
As `x in R`
`therefore (-1)^(2)-4(y)(y) ge 0`
`rArr 1-4y^(2) ge 0`
`rArr y in [(-1)/(2),(1)/(2)]`
` therefore "Range" ="Codomain"=[(-1)/(2),(1)/(2)]`
So, f(x) is surjective.
Hence, f(x) is surjective but not injective.
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