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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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