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JEE Main & Advanced Mathematics Functions Question Bank Self Evaluation Test - Relation and Functions-II

  • question_answer
    If \[f(x)=\left\{ \begin{matrix}    {{x}^{3}}+1,x<0  \\    {{x}^{2}}+1,x\ge 0  \\ \end{matrix},g(x)=\left\{ \begin{matrix}    {{(x-1)}^{1/3}},x<1  \\    {{(x-1)}^{1/3}},x\ge 1  \\ \end{matrix}, \right. \right.\] Then (gof) (x) is equal to

    A) \[x,\forall x\in R\]

    B) \[x-1,\forall x\in R\]

    C) \[x+1,\forall x\in R\]

    D) None of these

    Correct Answer: A

    Solution :

    [a] Let\[x<0\]. \[\therefore (gof)(x)=g(f(x))=g({{x}^{3}}+1)={{[({{x}^{3}}+1)-1]}^{1/3}}\] \[(\because x<0\Rightarrow {{x}^{3}}+1<1)\] \[={{({{x}^{3}})}^{1/3}}=x\] Let \[x\ge 0.\] \[\therefore (gof)(x)=g(f(x))=g({{x}^{2}}+1)={{(({{x}^{3}}+1)-1)}^{1/2}}\] \[(\because x\ge 0\Rightarrow {{x}^{2}}+1\ge 1)\] \[={{({{x}^{2}})}^{1/2}}=\left| x \right|=x\] \[(\because x\ge 0)\] \[\therefore (gof)(x)=x\forall x\in R\]


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