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The function f:R->[-1/2,1/2] defined as ...

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

A

neither injective nor surjective.

B

invertible.

C

injective but not surjective.

D

Surjective but not injective.

Text Solution

Verified by Experts

The correct Answer is:
D
`f:R to [-(1)/(2),(1)/(2)], f(x)=(x)/(1+x^(2)) AA x in R`
For range of the function, `y=(x)/(1+x^(2))`
`implies yx^(2)-x+y=0`
Since x is real, `D ge 0`
`implies 1-4y^(2) ge 0`
`implies y in [-(1)/(2),(1)/(2)]`
Also, for `y=(1)/(4),(1)/(4)=(x)/(1+x^(2))`
`or x^(2)-4x+1=0,` which has two distinct roots.
Thus, `f(x)` is onto but not injective.
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