If-f-R-R-is-a-polynomial-function-which-satisfy-the-equation-f-f-x-f-x-12x-x-R-then-find-f-x-

Question Number 22133 by Tinkutara last updated on 11/Oct/17

If f : R^{+} \to R^{+} is a polynomial function

which satisfy the equation f (f (x)) +

f (x) = 12 x \forall x \in R^{+}, then find f (x) .

Commented by Rasheed.Sindhi last updated on 22/Oct/17

f (f (x)) + f (x) = 12 x

Case - 1 : f (x) is linear

Let f (x) = ax + b

f (f (x)) + f (x) = 12 x \Rightarrow

{a (ax + b) + b} + ax + b = 12 x

a^{2} x + ab + b + ax + b = 12 x

(a^{2} + a) x + ab + 2 b = 12 x

Comparing coefficients :

a^{2} + a = 12 \land ab + 2 b = 0

(a - 3) (a + 4) = 0 \land b (a + 2) = 0

(a = 3 ∣ a = - 4) \land (b = 0 ∣ a = - 2)

f (x) = 3 x, f (x) = - 4 x

If f (x) is linear then

f (x) = 3 x or f (x) = - 4 x

Answered by ajfour last updated on 14/Oct/17

l e t f (x) = y

A s f (y) + f (x) = 12 x

\Rightarrow \frac{d f}{d y} \times f^{'} (x) + f^{'} (x) = 12

o r (\frac{d f}{d y} + 1) f^{'} (x) = 12

\Rightarrow \frac{d f}{d y} = c o n s t a n t & f^{'} (x) = c o n s t a n t

l e t f^{'} (x) = a, t h e n \frac{d f}{d y} = \frac{12}{a} - 1 . . (i)

\Rightarrow f^{'} (x) = a x + b (. . s a y)

f (y) = a y + b

a n d \frac{d f}{d y} = a \dots (i i)

f r o m (i) a n d (i i) :

a = \frac{12}{a} - 1 \Rightarrow a^{2} + a - 12 = 0

\Rightarrow a = - 4, 3

n o w f (y) + f (x) = 12 x

\Rightarrow (a y + b) + (a x + b) = 12 x

\Rightarrow [a (a x + b) + b] + a x + b = 12 x

o r (a^{2} + a) x + b (a + 2) = 12 x

\Rightarrow a^{2} + a = 12 a n d b (a + 2) = 0

S o a = - 4, 3 w h i l e b = 0

H e n c e f (x) = - 4 x

o r f (x) = 3 x .

Commented by Tinkutara last updated on 14/Oct/17

Thank you very much Sir!