Determine-f-x-when-f-f-x-x-

Question Number 1278 by Rasheed Soomro last updated on 19/Jul/15

D e t e r m i n e f (x) w h e n f (f (x)) = x .

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

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

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

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

Comparing coefficients

a + d = 0

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

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

More solution can be found .

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

f (x) = x

f (x) = k - x

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

Also more solutions are possible

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

Some solutions are given in comments

but more solutions exist .