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Let $f(x) = px + q$, where $p$ and $q$ are real numbers. Find $p+q$ if $f(f(f(x))) = 64x - 105 - 30x + 70 + 32x - 12$.

 Mar 2, 2024
 #1
avatar+128794 
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f (f ( f(x)))  =  66x - 47

 

f (f (x) ) =  p(px + q) + q =  p^2x + pq + q

 

f ( f ( f (x)) )  =  p (p^2x + pq + q) + q =   p^3x + p^2q + pq + q

 

So

 

p^3x + p^2q + pq + q  =  66x - 47

 

p^3  = 66

p= 66^(1/3)

 

And

p^2q + pq + q  =  -47

66^(2/3)q + 66^(1/3)*q + q = -47

q (66^(2/3) + 66^(1/3) + 1)  =  -47

q  = -47 / ( 66^(2/3) + 66^(1/3) + 1)

 

p + q =     [ 66^(1/3) ( 66^(2/3) + 66^(1/3) + 1)  - 47] / [ 66^(2/3) +66^(1/3) + 1 ]  =

 

[ 66^(2/3) + 66^(1/3) + 19 ]  /  [66^(2/3) + 66^(1/3) + 1 ]  ≈  1.842..... 

 

cool cool cool

 Mar 2, 2024

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