Let $f(x)=x^2-2x$. What is the value of $f(f(f(f(f(f(-1))))))$?
We start from the inside and work our way out: $$f(-1)=(-1)^2-2(-1)=3.$$ Therefore $$f(f(f(f(f(f(-1))))))=f(f(f(f(f(3)))).$$ Now $f(3)=3^2-2\cdot3=3$. We can use that fact repeatedly to conclude  \begin{align*}
f(f(f(f(f(f(-1))))))&=f(f(f(f(f(3))))\\
&=f(f(f(f(3)))\\
& \vdots\\ &= f(3)=\boxed{3}.\end{align*}
3
Level 3
Algebra
Problem: Let $f(x)=x^2-2x$. What is the value of $f(f(f(f(f(f(-1))))))$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of f(f(f(f(f(f(-1)))))), we first need to evaluate f(-1). 

1. Calculate f(-1):
   f(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3.

2. Next, we evaluate f(3):
   f(3) = 3^2 - 2(3) = 9 - 6 = 3.

3. Now, we evaluate f(3) again:
   f(3) = 3 (as calculated previously).

4. Continuing, we evaluate f(3) once more:
   f(3) = 3.

5. We repeat the evaluation of f(3):
   f(3) = 3.

6. Finally, we evaluate f(3) one last time:
   f(3) = 3.

After performing these calculations, we see that f(f(f(f(f(f(-1)))))) simplifies to f(3), which consistently evaluates to 3. Thus, the final answer is 3.<|END_OF_LATENT|> The final answer is $3$.