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f(x)=x^2-2x How would you find f(f(f(f(f(f(-1))))))?

 Feb 20, 2019
 #1
avatar+5233 
+1

\(\text{a little toying with it shows that }\\ f^{\circ k}(x) = c,~\forall k >0\\ \text{here }f^{\circ k} \text{ means applying the function }f, ~k \text{ times, i.e.}\\ f^{\circ 2}(x) = f(f(x))\\ \text{I leave it to you to determine }c \text{ and convince yourself that above is true}\)

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 Feb 20, 2019
 #2
avatar+101872 
+3

f(x)=x^2-2x How would you find f(f(f(f(f(f(-1))))))  ?

 

We (seemingly) have six evaluations to make

 

(1)  f ( - 1) =  (-1)^2 -2(-1)  = 1 + 2  = 3

 

(2)   f ( f  (-1) ) =  f (3) =    (3)^2 - 2(3) =  9 - 6 =  3

 

(3)  f (  f ( f  (-1) ) ) =  f(3)  = 3

 

Note that this pattern continues....so

 

f(f(f(f(f(f(-1))))))  =  3

 

 

cool cool cool

 Feb 20, 2019

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