Problem
Let $f$ and $g$ be defined by the table to the right. Find the following. \[ \sqrt{f(-2)-f(-1)}-[g(1)]^{2}+f(-3) \div g(1) \cdot g(-2) \] \begin{tabular}{|r|r|r|} \hline$x$ & $f(x)$ & $g(x)$ \\ \hline-3 & 3 & 8 \\ \hline-2 & 0 & 9 \\ \hline-1 & -9 & 9 \\ \hline 0 & -9 & 0 \\ \hline 1 & -8 & -3 \\ \hline \end{tabular} \[ \sqrt{f(-2)-f(-1)}-[g(1)]^{2}+f(-3) \div g(1) \cdot g(-2)= \] (Simplify your answer.)
Solution
Step 1 ：From the table, we find that $f(-2) = 0$, $f(-1) = -9$, $g(1) = -3$, $f(-3) = 3$, and $g(-2) = 9$.
Step 2 ：Substitute these values into the expression: $\sqrt{f(-2)-f(-1)}-[g(1)]^{2}+f(-3) \div g(1) \cdot g(-2) = \sqrt{0-(-9)}-(-3)^{2}+3 \div -3 \cdot 9$
Step 3 ：Simplify the expression: $\sqrt{9}-9-3 \cdot 9 = 3-9-27$
Step 4 ：Further simplify the expression to get the final answer: $3-9-27 = -33+9 = -24$
Step 5 ：Final Answer: \(\boxed{-24}\)
