Define-a-function-f-R-R-such-that-f-f-x-x-2-x-1-for-all-real-x-Evaluate-f-0-

Question Number 4671 by 314159 last updated on 20/Feb/16

D e f i n e a f u n c t i o n f : R \to R s u c h t h a t f (f (x)) = x^{2} - x + 1

f o r a l l r e a l x . E v a l u a t e f (0) .

Answered by 123456 last updated on 20/Feb/16

f (f (x)) = x^{2} - x + 1

f (f (f (x))) = f {(x)}^{2} - f (x) + 1

f (f (0)) = 1

f (f (1)) = 1

f (f (f (1))) = f {(1)}^{2} - f (1) + 1

f (1) = f {(1)}^{2} - f (1) + 1

f {(1)}^{2} - 2 f (1) + 1 = 0

{(f (1) - 1)}^{2} = 0

f (1) = 1

f (f (f (0))) = f {(0)}^{2} - f (0) + 1

f (1) = f {(0)}^{2} - f (0) + 1

1 = f {(0)}^{2} - f (0) + 1

f (0) (f (0) - 1) = 0

f (0) = 0 \lor f (0) = 1

if f (0) = 0

f (f (0)) = f (0) = 0 \neq 1

so

f (0) = 1

⋮

Commented by 123456 last updated on 21/Feb/16

f (0) = a \Rightarrow f (f (0)) = f (a) = 1

f (1) = b \Rightarrow f (f (1)) = f (b) = 1

f (f (a)) = f (1) = b

f (f (b)) = f (1) = b

a^{2} - a + 1 = b \Rightarrow a = 0 \lor a = 1

b^{2} - b + 1 = b \Rightarrow b = 1

f (x) = x

f (f (x)) = f (x) = x

x^{2} - x + 1 = x

x = 1

Commented by 314159 last updated on 21/Feb/16

T h a n k s a l o t!

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