Not double, but triple repeated functions...

Algebra Level 3

If f ( x ) = 5 x − 2 f(x) = 5x - 2 , then find the value of x x which satisfies f 3 ( x ) = − 20 − x f^3(x) = -20 - x . If x x can be expressed as an irreducible fraction a b \frac{a}{b} , find a + b a+b assuming a , b > 0 a, b>0 .

Note: f 3 ( x ) = f ( f ( f ( x ) ) ) f^3(x) = f(f(f(x))) , NOT ( f ( x ) ) 3 (f(x))^3 .


The answer is 4.

Noel Lo by Noel Lo , 24, Singapore

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1 solution

Noel Lo
May 10, 2015

f 3 ( x ) = f 2 ( f ( x ) ) = f 2 ( 5 x − 2 ) = f ( f ( 5 x − 2 ) ) f^3(x) = f^2(f(x)) = f^2(5x - 2) = f(f(5x - 2))

= f ( 5 ( 5 x − 2 ) − 2 ) = f ( 25 x − 10 − 2 ) = f ( 25 x − 12 ) = f(5(5x - 2) -2)= f(25x -10 -2) = f(25x -12)

= 5 ( 25 x − 12 ) − 2 = 125 x − 60 − 2 = 125 x − 62 =5(25x -12) -2 = 125x -60 -2 = 125x -62

125 x − 62 = − 20 − x 125x -62 = -20 -x

( 125 + 1 ) x = 62 − 20 (125+1)x =62-20

126 x = 42 126x = 42

x = 42 126 = 1 3 x=\frac{42}{126} =\frac{1}{3} so a + b = 1 + 3 = 4 a+b = 1+3 =\boxed{4}

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