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Find the range of f(x)=cos^(-1)((sqrt(1+...

Find the range of `f(x)=cos^(-1)((sqrt(1+2x^2))/1)`

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`f(x)="cos"^(-1)((sqrt(1+2x^(2)))/(1+x^(2)))`
`="sin"^(-1)(sqrt(1-(1+2x^(2))/((1+x^(2))^(2))))( :' cos^(-1)x=sin^(-1)sqrt(1-x^(2)))`
`="sin"^(-1)(sqrt((x^(4))/((1+x^(2))^(2))))`
`="sin"^(-1)((x^(2))/(1+x^(2)))`
`="sin"^(-1)(1-(1)/(1+x^(2)))`
Now, `(1-(1)/(1+x^(2))) in [0,1)`
` :. f(x) in [0,(pi)/(2))`
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