The range of the function `f(x)=(x)/(1+x^(2))` isA. `[0,1//2]`B. `[-1//2,1//2]`C. `[-1//2,0]`D. `[-1//2,0) cup (0,1//2]`

The range of the function `f(x)=(x)/(1+x^(2))` isA. `[0,1//2]`B. `[-1/

1.	The range of the function `f(x)=(x)/(1+x^(2))` isA. `[0,1//2]`B. `[-1//2,1//2]`C. `[-1//2,0]`D. `[-1//2,0) cup (0,1//2]`
Answer» Correct Answer - B Clearly, f(x) is defined for all ` x in R. ` So, domain (f)=R In order to find the range of f(x) , let y=f(x) `implies y=(x)/(x+x^(2)` `implies x^(2)y-x+y=0` `implies x=(1 pm sqrt(1-4y^(2)))/(2y)` For x to be real, we must have `1-4y^(2) ge 0 and y ne 0 implies -(1)/(2) le y le (1)/(2) and y ne 0` Also , y=-0 for x=0 Hence, range of f(x)=[-1/2,1/2]

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