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For the function f(x) = (ax + b)/(cx + d), the composition f(f(x)) equals x when:

1. d = a
2. a = b = c = d = 1
3. a = b = 1
4. d = -a

Correct answer: d = -a

Solution

For f(x)=(ax+b)/(cx+d), f is an involution (f(f(x))=x) exactly when its trace vanishes, a+d=0, i.e. d=-a. Substituting d=-a indeed gives f(f(x))=x, whereas a=b=c=d=1 gives the constant 1.

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