Problem:
Let f be the function defined by f(x)=ax2−2​ for some positive a. If f(f(2​))=−2​ then a=
Answer Choices:
A. 22−2​​
B. 21​
C. 2−2​
D. 22​​
E. 22+2​​
Solution:
Since aî€ =0, the only x for which f(x)=−2​ is x=0. Since f(f(2​))=−2​,f(2​) must be 0. Thus 2a−2​=0, or a=22​​.
OR
Since f(2​)=2a−2​,f(f(2​))=a(2a−2​)2−2​ which we set equal to −2​. Therefore, a(2a−2​)2=0. Since a>0,2a=2​ and a=22​​.