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Let g: Rvec(0,pi/3) be defined by g(x)=c...

Let `g: Rvec(0,pi/3)` be defined by `g(x)=cos^(-1)((x^2-k)/(1+x^2))` . Then find the possible values of `k` for which `g` is a subjective function.

Text Solution

Verified by Experts

The correct Answer is:
(-1, -1/2]
`g(x)` is surjective if
`(1)/(2) le (x^(2)-k)/(1+x^(2)) lt AA x in R`
or `(1)/(2) le 1-((k+1))/(x^(2)+1) lt 1 AA x in R`
or `-(1)/(2) le 1-((k+1))/(x^(2)+1) lt 0 AA x in R`
or `0 lt ((k+1))/(x^(2)+1) le (1)/(2) AA x in R`
or `k+1 gt -1,`
So, `k gt -1 " (1)" `
and `(k+1)/(x^(2)+1) le (1)/(2) AA x in R`
or `x^(2)-(2k+1) ge 0 AA x in R`
or `4(2k+1) le 0`
` :. k le -(1)/(2) " (2) " `
From (1) and (2) , `k in (-1,-(1)/(2)].`
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