f-1-f-x-f-x-f-x-

Question Number 1280 by Rasheed Soomro last updated on 19/Jul/15

f (\frac{1}{f (x)}) = f (x)

f (x) = ?

Commented by 123456 last updated on 19/Jul/15

f (x) = \frac{1}{x}

Commented by prakash jain last updated on 19/Jul/15

f (x) = \frac{a x + b}{c x + d}

f (\frac{1}{f (x)}) = \frac{a \frac{c x + d}{a x + b} + b}{c \frac{c x + d}{a x + b} + d} = \frac{a x + b}{c x + d}

[a (b + c) x + (b^{2} + a d)] (c x + d)] = [(c^{2} + a d) x + d (b + c)] (a x + b)

a c (b + c) = a (c^{2} + a d) \Rightarrow b c = a d

b^{2} c + a c d + a b d + a c d = a d (b + c) + (c^{2} + a d) b

\Rightarrow b^{2} c + a c d = c^{2} b + a b d \Rightarrow b c = a d

d (b^{2} + a d) = b d (b + c)

\Rightarrow {d b}^{2} + {a d}^{2} = b^{2} d + b c d \Rightarrow b c = a d

f (x) = \frac{a x + b}{c x + \frac{b c}{a}} = \frac{a^{2} x + a b}{a c x + b c}

This functional equation will also

have multiple solutions .