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Practice Paper 1 Question 2

Let \(f(x)\) mean that the function \(f\) is applied to \(x\), and \(f^n(x)\) mean \(f(f(...f(x)))\), that is \(f\) is applied to \(x\), \(n\) times. Let \(g(x)=x+1\) and \(h_n(x)=g^n(x)\). What is \(h_n^m(0)\)?

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Hints

  • Hint 1
    What is \(g^n(x)\) in terms of \(x\) and \(n,\) and how does that relate to the function \(h\)?
  • Hint 2
    How about trying to first find an expression \(h^2_n(x)\)?
  • Hint 3
    By trying values of \(m,\) give an expression for \(g_n^m(x).\)

Solution

We have \(h_n(x)=g^n(x)=x+n\) since \(1\) must be added \(n\) times. To compute \(h_n^m(0)\) we may first obtain \(h_n^m(x),\) and then evaluate it at \(x=0.\) Consider that \(h_n^2(x)=h_n(h_n(x))=h_n(x+n)=x+2n.\) We can hence inductively see that \(h_n^m(x)=x+mn.\) Therefore \(h_n^m(0)=mn.\)

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