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For some constants $a$ and $b$ let

\[f(x) = \left\{
\begin{array}{cl}
9 - 2x & \text{if } x \le 3, \\
ax + b & \text{if } x > 3.
\end{array}
\right.\]

The function $f$ has the property that $f(f(x)) = x$ for all $x$. What is $a+b$?

 Apr 17, 2021
 #1
avatar+420 
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if \(f(f(x)) = x\) for all x, then f(x) is an involution. The properties of an involution include the fact that it is reflexive across the line y=x, which means that y=ax+b must be a reflected version of y=9-2x through the line y=x. To reflect that line, just find the inverse of that line by changing y for x and solving for y:

\(x=9-2y\\2y=9-x\\y=4.5-0.5x\)

a = -0.5, b = 4.5, so a+b \(\boxed{4}\)

 Apr 17, 2021

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