The table provided involves completing function compositions given two one-to-one functions $f(x)$ and $g(x)$. Let's carefully analyze how to fill in the remaining blanks in the table.

Definitions:

• $(f \circ g)(x) = f(g(x))$ means you apply $g(x)$ first and then apply $f$ to the result.
• $(g \circ f)(x) = g(f(x))$ means you apply $f(x)$ first and then apply $g$ to the result.

Information from the table:

• $f(-5) = -3$, $f(-3) = 13$, $f(13) = 9$, and $f(9)$ is missing.
• $g(-5) = -5$, $g(-3) = 9$, $g(9) = -3$, and $g(13)$ is missing.

We are given some values of the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$, but several values are still missing. Let’s use the known values to solve for the unknowns.

Step-by-step completion of missing values:

1. $g(13)$:

   • We are given $(f \circ g)(13) = 9$, so $f(g(13)) = 9$.
   • Since $f(9) = 9$ (from the table), $g(13) = 9$.

2. $f(9)$:

   • From $(f \circ g)(9) = 13$, we know $f(g(9)) = 13$.
   • Since $g(9) = -3$, $f(-3) = 13$, which confirms the existing values.
   • Therefore, $f(9) = 13$.

3. $(g \circ f)(-3)$:

   • We are given $f(-3) = 13$. Now, $g(f(-3)) = g(13) = 9$.
   • So, $(g \circ f)(-3) = 9$.

Thus, the final completed table looks like this:

Would you like more details or further clarification on any step?

Here are 5 related questions:

1. What does it mean for a function to be "one-to-one"?
2. How would the table change if the functions were not one-to-one?
3. Can you determine the inverse of the function compositions from this table?
4. How would you represent these function compositions graphically?
5. What additional information could be inferred if the functions were provided as algebraic expressions?

Tip: When working with function compositions, always apply the innermost function first before moving outward.