Let f be the function defined by  f(x)=ax^2-(the square root of 2) for
some positive a.  If  f(f(the square root of 2)) = -(the square root
of 2),
find a. (show work)

Hi again!!

I will use the following notation:
the square root of 2 = sqrt(2)


f(x) = a.x^2 - sqrt(2)

then:

f(sqrt(2)) = a.(sqrt(2))^2 - sqrt(2) =
           = 2.a - sqrt(2)

f(f(sqrt(2))) = f(2.a - sqrt(2)) =
              = a.(2.a - sqrt(2))^2 - sqrt(2)

f(f(sqrt(2))) = - sqrt(2)

Then:
a.(2.a - sqrt(2))^2 - sqrt(2) = - sqrt(2) ==>

==> a.(2.a - sqrt(2))^2 = 0 ==> (a > 0)

==> (2.a - sqrt(2))^2 = 0 ==>

==> 2.a - sqrt(2) = 0 ==>

==> 2.a = sqrt(2) ==>

==> a = sqrt(2)/2 = sqrt(1/2) = 1/sqrt(2)


Note that now we can see that f(sqrt(2)) = 0 , then
f(f(sqrt(2)) = f(0) = -sqrt(2)


I hope this helps you.

Regards.
livioflores-ga