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BCECE Engineering BCECE Engineering Solved Paper-2014

  • question_answer
    The function \[f:R\to R\] is defined by \[f\left( x \right)={{3}^{-x}}\]. Observe the following statements I. \[f\] is one-one. II. \[f\] is onto. III.\[f\] s a decreasing function. Out of these, true statements are

    A) I, III           

    B) II, Ill

    C)  l, ll                         

    D) All of these    

    Correct Answer: A

    Solution :

    Given, \[f:R\to R\]such that \[f(x)={{3}^{-x}}\]Let \[{{y}_{1}}\]and \[{{y}_{2}}\]be two elements of \[f(x)\]such that \[{{y}_{1}}={{y}_{2}}\] \[\Rightarrow {{3}^{-{{x}_{1}}}}={{3}^{-{{x}_{2}}}}\Rightarrow {{x}_{1}}={{x}_{2}}\] So, \[f(x)\]is one ? one. Since, \[f(x)\] is positive for every value of \[x,\]therefore \[f(x)\]is into. Now, \[f(x)={{3}^{-x}}\] \[f'(x)=-{{3}^{-x}}\log 3<0\forall x\in R\] \[\therefore \]it  is decreasing function. Hence, statements I and III are true.                


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