Stuck on this function

[Jeff Ford]

I'm going through the review section of the Stewart Calculus text and I'm stuck on this problem.

Given
$f_0(x) = x^2$
and
$f_0(f_n(x)) = f_{n+1}(x) , n = 0,1,2...$
how do you solve for
$f_n(x)$

 

[Jeff Ford]

The only place I can think to start is making $(f_n(x))^2 = f_{n+1}(x)$, but that's as far as I got

 

[VietDao29]

Now, f₁(x) = f₀(f₀(x)) = f₀(x²) = x⁴
f₂(x) = f₀(f₁(x)) = f₀(x⁴) = x⁸
f₃(x) = f₀(f₂(x)) = ...
f₄(x) = f₀(f₃(x)) = ...
So what's f_n(x)?
Can you go from here?
Viet Dao,

 

[Jeff Ford]

From that I get $f_n(x) = x^{2^{n+1}}$ but that doesn't work for $f_0(x) = x^2$. That's the same answer the book has, so maybe I wrote the requirements down wrong. I'll have to check when I get home if this had to work for 0.

 

[Jeff Ford]

Yes it does. Never mind.

Thanks Viet Dao!