2018 in Algebra (1)

Algebra Level 3

Let $f$ be the function such that:

$\large \begin{cases} 2x & \text{ if } x\le \frac{1}{2} \\ 2-2x & \text{ if } x > \frac{1}{2} \end{cases}$

What is the total length of the graph of $\underbrace{f(f(\dots f(x)}_{2018 f's}\dots))$ from $x=0$ to $x=1?$

\sqrt {4^{2018} +1}

\sqrt {4^{2017} +2}

\sqrt {2^{2018} +1}

\sqrt {4^{4036} +1}

\sqrt {4^{2018} +2018}

1 solution

Hana Wehbi
Feb 15, 2018

When there are $n$ copies of $f$ , the graph consists of $2^n$ segments, each of which goes to $\frac{1}{2^n}$ units to the right, and alternately $1$ unit up or down. So the length is:

$2^n\sqrt{1+\frac{1}{2^n}} = \sqrt{4^n+1}$ ,

Taking $n=2018$ , the answer is $\sqrt {4^{2018} +1}$ .

2018 in Algebra (1)

1 solution

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