algebra 2: functions?

Question

Given that f(x) is a linear function,
and g(x) =x^2
find f(x), if
g(f(f(x))) = 9x^2 + (6pi squareroot3 +6pi)x +4pi^2 + 2squareoot3 pi^2.

sorry i cant make symbols >_<. help is much appreciated.

Answer 1

g(f(f(x))) = 9x^2 + (6π√3 + 6π)x +4π^2 + 2√3 π^2.

To solve this, work it from the outside in by undoing the operations. Thus, if g(x) = x^2, to undo the g(x) function, take the square root.

f(f(x)) = √(9x^2 + (6π√3 + 6π)x +4π^2 + 2√3 π^2)

Redistribute back into parenthesis to form a quadratic:
9x^2 + (6π√3 + 6π)x + (4 + 2√3)π^2

Remember, when squaring a monomial of the form ax + b:
a^2x^2 + 2abx + b^2 = (ax + b)^2

The first term is 9x^2, so a = 3.

The middle term is 2abx, so plug a in:
2(3)bx = 6bx
6bx = (6π√3 + 6π)x
b = π√3 + π

Now that we have a and b, that gives us:
f(f(x)) = ax + b = 3x + π√3 + π

Now, to undo f(x), there's an easy way:
f(x) is a linear function, of the sort ax + b.
If f(x) = ax + b, then f(f(x)) = a(ax + b) + b = a^2x + ab + b.

Taking our formula 3x + π√3 + π, it's obvious that b is π, so a must be √3.
That means f(x) = x√3 + π (solution!)

Answer 2

I don't know exactly how to work that problem BUT, I can show you how to make the symbols...^_^.

All you have to do is press Alt and then press a number. The numbers you press must be from the numlock pad... So,

Alt+253 ²
Alt+251 √
Alt+227 π
Alt+0179 ³
Alt+3 ♥♥

Good Luck with that problem...=).