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JEE Main & Advanced Mathematics Functions Question Bank Self Evaluation Test - Relations and Functions-I

  • question_answer
    A real valued function f(x) satisfies the functional equation \[f(x-y)=f(x)f(y)-f(a-x)f(a+y)\]where a is a given constant and \[f(0)=1,f(2a-x)\] is equal to

    A) \[-f(x)\]

    B)        \[f(x)\]

    C) \[f(a)+f(a-x)\]

    D)        \[f(-x)\]

    Correct Answer: A

    Solution :

    [a] \[f(2a-x)=f(a-(x-a))=f(a)f(x-a)-f(0)\] \[f(x)=f(a)f(x-a)-f(x)=-f(x)\] \[[\because \,\,\,x=0,y=0,f(0)={{f}^{2}}(0)-{{f}^{2}}(a)\] \[\Rightarrow {{f}^{2}}(a)=0\] \[\Rightarrow f(a)=0]\Rightarrow f(2a-x)=-f(x)\]


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