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For some constants a and b let

f(x) = 8 - x if x <= 4

f(x) = ax + b if x > 4

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

 May 13, 2022
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Note that f(x) will always put out a value greater than or equal to four for an imput of less than or equal to four. 

 

For x<=4:

f(f(x)) = a(f(x))+b = a(8 - x) + b = x

 

We want (a,b) such that a(8 - x) + b = x for x<=4.

-ax + 8a + b = x

 

There is only one term on the left that contains x, which means that it must be equal to x. The only value of a that would do that is a = -1.

 

x - 8 + b = x

b = 8

 

a + b = 7

 

(I didn't check this, it could be wrong)

 May 13, 2022

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