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KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    Let \[f:R\to R\] and \[g:R\to R\] be two one-one and onto functions such that they are the mirror images of each other about the line \[y=a.\] If \[h(x)=f(x)+\text{ }g(x),\] then \[h(x)\] is

    A) one-one onto

    B) one-one into

    C) many-one onto

    D) many-one into

    Correct Answer: D

    Solution :

    Since \[f(x)\] and \[g(x)\]are mirror images of each other about the line\[y=a,\]\[f(x)\]and \[g(x)\]are at equal distances from the line \[y=a\], Let for some particular \[{{x}_{0}}\]
    \[f\left( {{x}_{0}} \right)=a+k,\operatorname{then}\,g\text{ }\left( {{x}_{0}} \right)=a-k,\]then \[h\left( {{x}_{0}} \right)=f\left( {{x}_{0}} \right)+g\left( {{x}_{0}} \right)=2a\]
    \[\therefore h\left( x \right)=2a\,\,\forall \,\,x\in R.\]So,\[h(x)\]  must be a constant function, which is many -one into.


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