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# Challenging algebra

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For some constants a and b let

$$f(x) = \left\{ \begin{array}{cl} ax + b & \text{if } x < 2, \\ 8 - 3x & \text{if } x \ge 2. \end{array} \right.$$

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

Aug 22, 2022

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a(8-3x)+b=x

-3ax+8a+b=x

a would need to be -1/3 in order to keep the x term on the left isolated. 8a+b would need to be equal to 0, so be would be 8/3. 8/3+(-1/3)=7/3.

Aug 23, 2022