Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right)$.
If $f(2)=5$ and $f(3)=4$, then $f^{-1}(5)=2$ and $f^{-1}(4)=3$, respectively. Therefore, $f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right)=f^{-1}\left(2+3\right)=f^{-1}(5) = \boxed{2}$.
2
Level 4
Algebra
Problem: Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right)$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve for \( f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right) \), we first need to determine the values of \( f^{-1}(5) \) and \( f^{-1}(4) \).

From the given information:
- \( f(2) = 5 \) implies \( f^{-1}(5) = 2 \).
- \( f(3) = 4 \) implies \( f^{-1}(4) = 3 \).

Now we can substitute these values into the expression:
- Calculate \( f^{-1}(5) + f^{-1}(4) \):
  \[
  f^{-1}(5) + f^{-1}(4) = 2 + 3 = 5.
  \]

Next, we need to evaluate \( f^{-1}(5) \):
- We already established that \( f^{-1}(5) = 2 \).

Thus, we find:
\[
f^{-1}\left(f^{-1}(5) + f^{-1}(4)\right) = f^{-1}(5) = 2.
\]

The final answer is \( 2 \).<|END_OF_LATENT|> The final answer is $2$.