Problem:
If the function f defined by
f(x)=2x+3cxβ,xξ =β23β,
satisfies f(f(x))=x for all real numbers x except β23β, then c is
Answer Choices:
A. β3
B. β23β
C. 23β
D. 3
E. not uniquely determined by the given information
Solution:
For all xξ =β23β,
x=f(f(x))=2(2x+3cxβ)+3c(2x+3cxβ)β=2cx+6x+9c2xβ
which implies (2c+6)x+(9βc2)=0. Therefore, 2c+6=0 and 9βc2=0. Thus, c=β3.