Question

Let $g:R\to (0,\frac{\pi }{3})$ be defined by $g\left(x\right)={\cos }^{-1}\left(\frac{x^2-k}{1+x^2}\right)$ Then find the possible values of $k$ for which $g$ is a subjective function

Answer 1

for subjective function co-domain= Range

so range of function will be $(0,\frac{\pi }{3})$

$0<{\cos }^{-1}\left(\frac{x^2-k}{1+x^2}\right)<\frac{\pi }{3}$

$\frac{1}{2}<\frac{x^2-k}{1+x^2}<1$

so we take first $\frac{x^2-k}{1+x^2}<1$

we get $k>-1$

than we take $\frac{1}{2}<\frac{x^2-k}{1+x^2}$

in this we get $k<\frac{1}{2}$

so we will take intersection both inequalities

and we get range of k is $kϵ(-1,\frac{1}{2})$.