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I've been working on this problem a while and I'm quite stuck. Could somebody please help me?

 

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 \(l\) be the length of the graph of \(y=k(x)\) for \(-3.5\le x\le 3.5\). What is the value of \(\ell^2\)?

 

 May 15, 2024

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