1. The problem states that $f$ is a one-to-one function with $f(-5) = -11$ and $f(2) = -12$. 2. Recall the definition of an inverse function: if $f(a) = b$, then $f^{-1}(b) = a$. 3. Using this, from $f(-5) = -11$, we get $f^{-1}(-11) = -5$. 4. From $f(2) = -12$, we get $f^{-1}(-12) = 2$. 5. Now, check each option: - $f^{-1}(-12) = -5$ is false because $f^{-1}(-12) = 2$. - $f^{-1}(-11) = -5$ is true. - $f^{-1}(-5) = 11$ is false; $-5$ is an input to $f$, not an output. - $f^{-1}(-11) = -12$ is false; $f^{-1}(-11) = -5$. Final answer: Only $f^{-1}(-11) = -5$ must be true.