Let's go through the table step by step to complete it, given that $f$ and $g$ are one-to-one functions.

Step 1: Analyze Given Information

• Given values in the table:
  • $f(-10) = 10$
  • $f(-3) = -3$
  • $f(10) = -10$
  • $g(-10) = 13$
  • $g(10) = -10$
  • $(f \circ g)(10) = -3$
  • $(g \circ f)(-10) = -10$
  • $(g \circ f)(10) = -3$

Step 2: Use the One-to-One Property

• Using $f(-10) = 10$:

  • If $f(-10) = 10$, then $g(10) = -10$.

• Using $f(-3) = -3$:

  • If $f(-3) = -3$, and $g$ is one-to-one, then $g(-3)$ should have a unique corresponding value in the set of $x$.

• Using $f(10) = -10$:

  • If $f(10) = -10$, and $g$ is one-to-one, then $g(-10) = 13$.

Step 3: Complete $g(x)$ values

• $g(13) = x$ (This is still unknown, but we can determine this value by seeing what value hasn't been used yet in the set).

Step 4: Fill in Compositions $(f \circ g)(x)$ and $(g \circ f)(x)$

• For $x = -10$:

  • $(f \circ g)(-10) = f(g(-10)) = f(13)$.
  • Since $f(13)$ is not yet known, leave this blank for now.
  • $(g \circ f)(-10) = g(f(-10)) = g(10) = -10$.

• For $x = -3$:

  • $(f \circ g)(-3) = f(g(-3))$.
  • $(g \circ f)(-3) = g(f(-3)) = g(-3)$.

• For $x = 10$:

  • $(f \circ g)(10) = f(g(10)) = f(-10) = 10$.
  • $(g \circ f)(10) = g(f(10)) = g(-10) = 13$.

Step 5: Determine Missing Values

Given the one-to-one property:

• $g(13) = -3$ (because no other values match and $f(13) = -10$).
• $f(13) = -10$ based on $(f \circ g)(13) = -10$.

Final Table:

$x$	$-10$	$-3$	$10$	$13$
$f(x)$	10	$-3$	$-10$	-3
$g(x)$	13	10	-10	-3
$f \circ g(x)$	13	-10	10	-3
$g \circ f(x)$	-10	-3	-3	13

Would you like to go over any specific part of this table in detail, or do you have any further questions?

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Follow-up Questions:

1. How does the one-to-one property help in solving problems like this?
2. What are the possible values for $f(g(x))$ if $g(x) = 13$?
3. Can you explain why the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$ might be useful in function analysis?
4. How would you approach this problem if $f$ and $g$ were not one-to-one functions?
5. What would change in the table if $g(13)$ had been different?

Tip:

Always cross-check the values by substituting back into the functions, as it can help to verify the correctness of your work.