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Question Number 24764 by NECx last updated on 25/Nov/17

G i v e n t h a t t h e f u n c t i o n f : R \to R

i s d e f i n e d b y f (x) = x^{n} . F o r w h a t

v a l u e s o f n, i f a n y, i s f o f = f . f ?

F o r e a c h o f t h e s e v a l u e s o f n f i n d

f o f .

Answered by mrW1 last updated on 25/Nov/17

{(x^{n})}^{n} = x^{n} \times x^{n}

x^{n \times n} = x^{n + n}

\Rightarrow n \times n = n + n

\Rightarrow n = 0 o r 2

w i t h n = 0

f (x) = 1

f o f = 1

w i t h n = 2

f (x) = x^{2}

f o f = x^{4}

Answered by ajfour last updated on 25/Nov/17

f o f = f [f (x)] = f (x^{n}) = {(x^{n})}^{n} = x^{(n^{2})}

f . f = {(x^{n})}^{2} = x^{2 n}

f o f = f . f \Rightarrow n^{2} = 2 n

o r n (n - 2) = 0 \Rightarrow n = 0, 2

f o r n = 0 :

f (x) = x^{0} = 1 h e n c e f o f = 1

f o r n = 2 :

f (x) = x^{2} h e n c e f o f = x^{4} .

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