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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 $\ell^2$?

 

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

 Dec 7, 2020
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The graph of y = k(x) is the inverse of the graph of j(x), due to the min/max relationship.  So we can invert the graph of the length of the graph of to find the length of the graph of j(x)

 

Answer: l^2 = 24

 Dec 8, 2020

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