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The complete graph of $y=f(x),$ which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1.$) Let $a$ and $b$ be the largest negative integer and the smallest positive integer, respectively, such that the functions $g(x)=f(x)+ax$ and $h(x)=f(x)+bx$ are invertible. What is $a^2+b^2?$

 Nov 16, 2019
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The largest slope is 3 and the smallest slope is -4, so the answer is 3^2 + (-4)^2 = 25.

 Nov 16, 2019

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