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JEE Main & Advanced Mathematics Functions Question Bank Functions

  • question_answer
    Let \[g(x)=1+x-[x]\] and \[f(x)=\left\{ \begin{align}   & -1,\ x<0 \\  & 0,\ \ x=0,\  \\  & \text{1,}\ \ \ \text{x}>\text{0} \\ \end{align} \right.\]then for all \[x,\ f(g(x))\] is equal to [IIT Screening 2001; UPSEAT 2001]

    A)                    x

    B)            1

    C)                    \[f(x)\]

    D)            \[g(x)\]

    Correct Answer: B

    Solution :

               Here \[g(x)=1+n-n=1,\,x=n\in Z\]            \[1+n+k-n=1+k\], \[x=n+k\] (where \[n\in Z,\,0<k<1\])            Now \[f(g(x))=\left\{ \begin{align}   & -1,\,\,\,\,\,g(x)<0 \\  & \,\,\,0,\,\,\,\,g(x)=0 \\  & \,\,\,1,\,\,\,\,\,g(x)>0 \\ \end{align} \right.\]                    Clearly, \[g(x)>0\] for all x. So, \[f(g(x))=1\] for all x.


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