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Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \max\{f(x),g(x),h(x)\},$$$$k(x) = \min\{f(x),g(x),h(x)\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)$ is the least of these three values.  Shown below is the graph of $y=j(x)$ for $-3.5\le x\le 3.5$.  Let $\ell$ be the length of the graph of $y=k(x)$ for $-3.5\le x\le 3.5$. What is the value of $\ell^2$?

 

here is the graph:

 

https://latex.artofproblemsolving.com/4/e/b/4eb6df592ed8659cbdc2a19e71139c1829c7b9ad.png

 Dec 6, 2020
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Go to your Art of Problem Solving message board to get help with this question.

Do not cheat and defeat the whole purpose of the AoPS learning site by asking people here to do it for you.

 Dec 6, 2020

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