SOLUTION: Let f be the function defined by f(x)=ax^2- sqrt2 for some positive a. If f(f(sqrt2))=-sqrt2, find the exact value of a.

Algebra ->  Functions -> SOLUTION: Let f be the function defined by f(x)=ax^2- sqrt2 for some positive a. If f(f(sqrt2))=-sqrt2, find the exact value of a.      Log On


   

Question 959135: Let f be the function defined by f(x)=ax^2- sqrt2 for some positive a. If f(f(sqrt2))=-sqrt2, find the exact value of a.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
looks like x = sqrt(2)/2 will do it.

f(x) is equal to ax^2 - sqrt(2)

what is the value of a when f(f(sqrt(2)) = -sqrt(2)?

if f(x) is equal to ax^2 - sqrt(2)), then f(sqrt(2)) is equal to a * sqrt(2)^2 - sqrt(2).

you are simply replacing x with sqrt(2).

f(sqrt(2)) is equal to a * sqrt(2)^2 - sqrt(2) which is equal to a * 2 - sqrt(2) which is equal to 2a - sqrt(2).

you have f(sqrt(2)) is equal to 2a - sqrt(2).

f(f(sqrt(2))) is equal to f(2a - sqrt(2)).

you are simply replacing f(sqrt(2)) with 2a - sqrt(2)).

f(2a - sqrt(2)) is equal to a * (2a - sqrt(2))^2 - sqrt(2)

you are simply replacing x in f(x) = a * x^2 - sqrt(2) with (2a - sqrt(2)).

since (2a - sqrt(2))^2 is equal to 4a^2 - 4a*sqrt(2) + 2, your equation of:

f(2a - sqrt(2)) is equal to a * (2a - sqrt(2))^2 - sqrt(2)becomes:

f(2a - sqrt(2)) is equal to a * (4a^2 - 4a*sqrt(2) + 2) - sqrt(2).

you know two things.
1. a has to be positive.
2. the end result has to be - sqrt(2).

this end result can only occur if a * (4a^2 - 4a*sqrt(2) + 2) is equal to 0.

since a has to be positive, this can only occur if 4a^2 - 4a*sqrt(2) + 2 is equal to 0.

since this is a quadratic equation, set it equal to 0 and factor it using the quadratic formula.

you will get a = sqrt(2)/2.

here's the graph of your final equation of y = a * (4a^2 - 4a*sqrt(2) + 2) - sqrt(2).

you can see that the graph intersect with the graph of the line y = -sqrt(2) in two places.

one place is when a = 0 but this is invalid because a has to be positive.

the other place is when a is equal to sqrt(2)/2

sqrt(2)/2 is equal to .7071 on the graph.

-sqrt(2) is equal to -1.414

the solution is at the point (x,y) = .7071,-1.414).

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