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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?

 

So I don't even know where to start with this one... can someone please explain how to do this?

 Jun 11, 2020
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Since you want  f( f(x) )  =  x, you'll want the first part to be the inverse of the second part.

 

Since the second part is  8 - 3x   --->     y  =  8 - 3x

(interchange both x and y)                     x  =  8 - 3y

(solve for y)                                        x - 8  =  -3y

                                                      (8 - x)/3  =  y

 

So:  ax + b  =  (8 - x)/3  =  8/3  - x/3  =  -(1/3)x + (8/3)

 Jun 11, 2020

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