Question

if f:R+->R+ is a polynominal function satisfying the equation f(f(x))=6x-f(x), then f(17)=?​

Answer 1

Step-by-step explanation:

Given if f:R+->R+ is a polynomial function satisfying the equation f(f(x))=6x - f(x), then f(17)=?

• Now f (f(x) = 6x – f(x)
• So f (f(x) is something of f(x)
• The only possibility is f(x) is a linear function.
• So f(x) = ax + b
• Therefore f(ax + b) = 6x – (ax + b)
• So f(ax + b) = (6 – a)x – b
•    [  Now in place of x we can put ax + b
•       So f(ax + b) = a(ax + b)  + b]
• So a (ax + b) + b = (6 – a)x – b
• So a^2 x + ab + b = (6 – a) x – b
• So a^2 = 6 – a
• So a^2 + a – 6 = 0
• So a^2 + 3a – 2a – 6 = 0
• Or a(a + 3) – 2(a + 3) = 0
• Or (a + 3) (a – 2) = 0
• Or a = - 3, 2
• Now for the constant term we have
• So ab + b = - b
• Or ab + 2b = 0
• Or b(a + 2) = 0
• Or b = 0
• Now f(x) = ax + b if b = 0,
• So f(x) = ax
• So possibilities will be  
• So f(x) = 2x if a = 2
• So f(17) = 2(17) = 34
• So f(x) = - 3x if a = - 3
• So f(17) - - 3(17) = - 51

Reference link will be

https://brainly.in/question/7780979