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JEE Main & Advanced JEE Main Paper (Held On 09-Jan-2019 Morning)

  • question_answer
    For \[x\in R\text{ }-\left\{ 0,\text{ }1 \right\},\text{ }let\text{ }{{f}_{1}}\left( x \right)\,\,=\,\,\frac{1}{x}\,\,,\text{ }{{f}_{2}}\left( x \right)=1-x\] and \[{{f}_{3}}\left( x \right)=\frac{1}{1-x}\] be three given functions.  If a function, J(x) satisfies \[\left( fz\circ \text{ }J\circ \,\,{{f}_{1}} \right)\left( x \right)=\text{ }{{f}_{3}}\text{ }\left( x \right)\] then J(x) is equal to: [JEE Main Online Paper (Held On 09-Jan-2019 Morning]

    A) \[\frac{1}{x}{{f}_{3}}\,(x)\]

    B) \[{{f}_{2}}\,(x)\]

    C) \[{{f}_{3}}\,(x)\]

    D) \[{{f}_{1}}\,(x)\]

    Correct Answer: C

    Solution :

    Given \[{{f}_{1}}\,(x)\,=\,\frac{1}{x},\,\,{{f}_{2}}\,(x)\,\,=\,\,1-x,\,\,{{f}_{3}}(x)\,=\,\,\frac{1}{1-x}\]x \[{{f}_{2}}\,(J({{f}_{1}}(x)))\,\,=\,\,{{f}_{3}}\,(x)\] \[{{f}_{2}}\,\left( J\left( \frac{1}{x} \right) \right)\,\,=\,\,\frac{1}{1-x}\] \[1-J\,\left( \frac{1}{x} \right)\,\,=\,\,\frac{1}{1-x}\] \[J\,\left( \frac{1}{x} \right)\,\,=\,1-\,\,\frac{1}{1-x}\] \[J\,\left( \frac{1}{x} \right)\,\,=\,\,\,\frac{-x}{1-x}\] \[J\,\left( \frac{1}{x} \right)\,\,=\,\,\,\frac{-x}{x-1}\] Let     \[t\,\,=\,\,\frac{1}{x}\] \[J(t)\,\,=\,\,\frac{1/t}{1/t-1}\] \[J(t)\,\,=\,\,\frac{1}{1-t}\] \[\therefore \,\,\,\,\,\,J(x)\,\,=\,\,\frac{1}{1-x}\] \[=\,\,{{f}_{3}}(x)\]


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