For which numbers a,b,c and d will the function f(x)=(ax+b)/(cx+d) satisfy f(f(x))=x for all x?

Juan Leonard

Juan Leonard

Answered question

2022-10-27

For which numbers a,b,c and d will the function f ( x ) = ( a x + b ) ( c x + d ) satisfy f ( f ( x ) ) = x for all x?

Answer & Explanation

Kash Osborn

Kash Osborn

Beginner2022-10-28Added 18 answers

Step 1
You say that you understand the composition part. Hence, you should know that the condition reduces to
C x 2 + ( D A ) x B = 0 ,
where C = a c + c d , D A = b c + d 2 a 2 c d and B = a b + d 2 .. This quadratic vanishes identically when the coefficients vanish.
Step 2
Thus, you must solve the system
C = 0 , D = A , B = 0 ,
which contains the four unknowns a,b,c,d. Thus there are infinitely many solutions as defined by that system.

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