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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    If \[f(x)=3x+10\], \[g(x)={{x}^{2}}-1\], then \[{{(fog)}^{-1}}\] is equal to                     [UPSEAT 2001]

    A) \[{{\left( \frac{x-7}{3} \right)}^{1/2}}\]

    B) \[{{\left( \frac{x+7}{3} \right)}^{1/2}}\]

    C) \[{{\left( \frac{x-3}{7} \right)}^{1/2}}\]

    D) \[{{\left( \frac{x+3}{7} \right)}^{1/2}}\]

    Correct Answer: A

    Solution :

    • \[f(x)=3x+10\] and \[g(x)={{x}^{2}}-1\]           
    • Þ \[f\,o\,g=f(g(x))=3(g(x))+10\] =\[3({{x}^{2}}-1)+10\]=\[3{{x}^{2}}+7\]                      .....(i)           
    • Let \[3{{x}^{2}}+7=y\] Þ \[3{{x}^{2}}=y-7\]           
    • Þ  \[{{x}^{2}}=\frac{y-7}{3}\Rightarrow x={{\left( \frac{y-7}{3} \right)}^{1/2}}\]           
    • We know that \[f(x)=y\], then \[x={{f}^{-1}}(y)\]           
    • so \[{{(fog)}^{-1}}={{\left( \frac{x-7}{3} \right)}^{1/2}}\].


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