Question

Cho f(x) = ax + b thỏa mãn \(f\left(f\left(f\left(0\right)\right)\right)=2\) và \(f\left(f\left(f\left(1\right)\right)\right)=29\). Tìm a,b ?

Answer 1

f(0) = a . 0 + b = b

f(f(0)) = f(b) = a . b + b = ab + b

f(f(f(0))) = f(ab + b) = a . (ab + b) + b = a²b + ab + b

f(1) = a . 1 + b = a + b

f(f(1)) = f(a + b) = a . (a + b) + b = a² + ab + b

f(f(f(1))) = f(a² + ab + b) = a . (a² + ab + b) + b = a³ + a²b + ab + b

a³ + a²b + ab + b = 29

a²b + ab + b = 2

=> (a³ + a²b + ab + b) - (a²b + ab + b) = 29 - 2

a³+ a²b + ab + b - a²b - ab - b = 27

a³ = 3³

a = 3