A composition problem!

Algebra Level 3

Let $f(x)=|x-2|$ and $g(x)=\underbrace{f(f(\cdots f(x) \cdots ))}_{n \text{ times}}$ . If the equation $g(x)=k$ , where $k \in (0,2)$ , has $8$ distinct solutions, then find $n$ .

10 2 5 6 3 4 8

1 solution

Sahil Goyat
Dec 28, 2019

Check this graph for a solution and a visual understanding of the problem. _ _ graph

Non-graph solution:

f(x)=|x-2|=k has two distinct positive solutions for all k in (0,2) when find f(f(x)) we substitute x with f(x) so gets two more solutions one positive and one negative similarly applying f another time will increase the solutions by two so applying it four times gives eight solutions.

this wont be the case for k outside the range as the function f returns two positive values for only this range it can easily be verified.so for other values the first f(x) will return two values but one will be negative(will not further provide any solution)and the positive value will be greater than original so no new solutions will appear.I gave the link to the graph so it can be easy to visualize after all the problems feels beautiful only when visualized in some way .hope it helped :)

Can you write a solution without the help of a graph? An algebraic method?

Using graph is considered as cheating according to me.

Vilakshan Gupta - 1 year, 5 months ago

The graph was to help having an insight.

Sahil Goyat - 1 year, 5 months ago

A composition problem!

1 solution

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