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A function f randomly assigns values to f(1),f(2),f(3), and f(4) from the set {1,2,3,4} with replacement. What is the probability that f(f(x))=f(f(f(x))) for each  in the set {1,2,3,4}? Express your answer as a common fraction.

 Feb 15, 2024
 #1
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So, here is my best shot:

Let \(f(f(x)=y\)  and  \(y\in {1,2,3,4}\)

 

Therefore:

\(f(f(x))=f(f(f(x)))\) becomes \(y=f(y)\)

 

This means we just need to find the probability of \(f(x)=x\) and the value of \(f(f(x))\) can be anything, it doesnt matter.
Thus, the probability of


 \(f(x)=x\) where \(x\in {1,2,3,4}\)
\(= \boxed{\frac{1}{4}}\)

hope its right ngl

 Feb 15, 2024

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