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National and Regional Contests
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MAA AMC
AMC 12/AHSME
1990 AMC 12/AHSME
12
Function
Function
Source:
April 4, 2006
function
Problem Statement
Let
f
f
f
be the function defined by
f
(
x
)
=
a
x
2
−
2
f(x)=ax^2-\sqrt2
f
(
x
)
=
a
x
2
−
2
for some positive
a
a
a
. If
f
(
f
(
2
)
)
=
−
2
f(f(\sqrt2 ))=-\sqrt 2
f
(
f
(
2
))
=
−
2
, then
a
=
a=
a
=
(A)
2
−
2
2
(B)
1
2
(C)
2
−
2
(D)
2
2
(E)
2
+
2
2
\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}
(A)
2
2
−
2
(B)
2
1
(C)
2
−
2
(D)
2
2
(E)
2
2
+
2
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