Problem:
If the function f defined by
f(x)=2x+3cx​,xî€ =−23​,
satisfies f(f(x))=x for all real numbers x except −23​, then c is
Answer Choices:
A. −3
B. −23​
C. 23​
D. 3
E. not uniquely determined by the given information
Solution:
For all xî€ =−23​,
x=f(f(x))=2(2x+3cx​)+3c(2x+3cx​)​=2cx+6x+9c2x​
which implies (2c+6)x+(9−c2)=0. Therefore, 2c+6=0 and 9−c2=0. Thus, c=−3.