1. The function f(x) = cot^-1[(x+3)x]^1/2+ cos^-1[x² +3x+1]^1/2is defined on the set S , where S equal to
(A) {0 , 3} (B) (0 ,3) (C) {0, -3} (D) [-3, 0]

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{The} \mathrm{given} \mathrm{function} \mathrm{is}, \\ \mathrm{f}\left(\mathrm{x}\right)={\cot }^{-1}\left({\left(\left(\mathrm{x}+3\right)\mathrm{x}\right)}^{\frac{1}{2}}\right)+{\cos }^{-1}\left({\left({\mathrm{x}}^2+3\mathrm{x}+1\right)}^{\frac{1}{2}}\right) \\ \mathrm{Note} \mathrm{that} \\ \mathrm{the} \mathrm{domain} \mathrm{of} {\cot }^{-1}\mathrm{x} \mathrm{is} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{real} \mathrm{numbers} \mathrm{and}, \\ \mathrm{the} \mathrm{domain} \mathrm{of} {\cos }^{-1}\mathrm{x} \mathrm{is} -1\le \mathrm{x}\le 1. \\ \mathrm{Also} \mathrm{note} \mathrm{that} {\mathrm{x}}^{\frac{1}{2}} \mathrm{is} \mathrm{defined} \mathrm{only} \mathrm{on} \mathrm{non}-\mathrm{negative} \mathrm{x}. \\ \mathrm{Hence} \left(\mathrm{x}+3\right)\mathrm{x} \mathrm{and} {\mathrm{x}}^2+3\mathrm{x}+1 \mathrm{should} \mathrm{be} \mathrm{non}-\mathrm{negative}. \\ \mathrm{It} \mathrm{can} \mathrm{be} \mathrm{ssen} \mathrm{that} \left(\mathrm{x}+3\right)\mathrm{x} \mathrm{is} \mathrm{negative} \mathrm{in} \left(-3, 0\right). \\ \mathrm{Then} \left(\mathrm{x}+3\right)\mathrm{x} \mathrm{is} \mathrm{positive} \mathrm{in} \mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{3}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{\infty }\mathbf{)}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }....\left(1\right) \\ \mathrm{Since} {\cos }^{-1}\mathrm{x} \mathrm{is} \mathrm{not} \mathrm{defined} \mathrm{for} \mathrm{x} \mathrm{greater} \mathrm{than} 1 \mathrm{and} \mathrm{less} \mathrm{than} -1, \\ -1\le {\left({\mathrm{x}}^2+3\mathrm{x}+1\right)}^{\frac{1}{2}}\le 1 \\ \mathrm{Or}, \\ {\mathrm{x}}^2+3\mathrm{x}+1\le 1 \\ {\mathrm{x}}^2+3\mathrm{x}\le 0 \\ \mathrm{x}\left(\mathrm{x}+3\right)\le 0 \\ \mathrm{This} \mathrm{inequality} \mathrm{is} \mathrm{valid} \mathrm{only} \mathrm{in} \left[\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{ }\mathbf{0}\right] ....\left(2\right) \\ \mathrm{Considering} \left(1\right) \mathrm{and} \left(2\right) \mathrm{together}, \mathrm{it} \mathrm{can} \mathrm{be} \mathrm{seen} \mathrm{that} \mathrm{both} \mathrm{the} \mathrm{conditions} \mathrm{are} \mathrm{satisfied} \mathrm{only} \mathrm{at} \mathrm{two} \mathrm{points}, \mathrm{at} -3 \mathrm{and} \mathrm{at} 0. \\ \mathrm{Hence} \mathrm{the} \mathrm{domain} \mathrm{is}, \left\{\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{ }\mathbf{0}\right\} \\ \mathrm{If} \mathrm{we} \mathrm{consider} \mathrm{the} \mathrm{argument} \mathrm{of} {\cos }^{-1}, \\ \mathrm{here} \mathrm{also} {\mathrm{x}}^2+3\mathrm{x}+1 \mathrm{should} \mathrm{be} \mathrm{non}-\mathrm{negative}. \\ \mathrm{In} \mathrm{the} \mathrm{domain} \left\{-3, 0\right\}, \mathrm{this} \mathrm{is} \mathrm{indeed} \mathrm{non}-\mathrm{negative}. \\ \mathrm{Hence} \mathrm{the} \mathrm{domain} \mathrm{remains} \mathrm{the} \mathrm{same} \mathrm{as} \left\{-3, 0\right\}. \end{gathered}$$

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