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Complete the table below. Assume 𝑓 and 𝑔 are one-to-one functions.

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.

Explore how to complete a table for one-to-one functions f and g with specific values provided. Understand the composition of functions and their properties. Suitable for high school students studying functions and algebra.

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