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The range of the function f(x)=(x)/(1+x^...

The range of the function `f(x)=(x)/(1+x^2),x in R ,` is

A

`(1,infty)`

B

`(-1/2,1/2)`

C

`(0,1/2)`

D

`(0,1)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `y=f(x)=(x^(2)+x+2)/(x^(2)+x+1), x in R`
`therefore y=(x^(2)+x+2)/(x^(2)+x+1)`
`y=1+(1)/(x^(2)+x+1) " "[i.e. y gt 1]` ...(i)
` rArr yx^(2)+yx+y=x^(2)+x+2`
` rArr x^(2) (y-1) + x(y-1)+(y-2)=0, AA x in R`
Since, x is real, `D ge 0`
`rArr (y-1)^(2)-4(y-1)(y-2) ge 0`
` rArr (y-1) {(y-1) -4(y-2)} ge 0`
`rArr (y-1)(-3y+7) ge 0`
`rArr 1 le y le (7)/(3) " ...(ii)" `
From Eqs. (i) and (ii), Range ` in (1,(7)/(3)]`
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