https://file.aiquickdraw.com/mathbot/file/1726443672468pphhbgjq.jpg

It seems the text extraction was not successful due to image quality or formatting. However, based on what is visible in the image, I can still assist with solving the problem.

### Given table:

| x  | y  |
|----|----|
| 3  | 15 |
| 7  | 19 |
| 9  | 21 |
| 8  | 20 |
| 45 | 45 |

#### Steps to find the rule:
1. Start by observing how \( y \) changes with respect to \( x \).
   - For \( x = 3 \), \( y = 15 \)
   - For \( x = 7 \), \( y = 19 \)
   - For \( x = 9 \), \( y = 21 \)
   - For \( x = 8 \), \( y = 20 \)
   - For \( x = 45 \), \( y = 45 \)

The difference between \( x \) and \( y \) is not linear. Instead, let's focus on individual differences. 

#### For smaller values of \( x \):
\[
y = x + 12
\]

However, when \( x = 45 \), \( y = 45 \), suggesting that for large values of \( x \), the rule becomes \( y = x \).

Thus, a possible piecewise function could be:

\[
y = \begin{cases} 
x + 12, & \text{if } x < 45 \\
x, & \text{if } x = 45 
\end{cases}
\]

The rule seems to depend on this pattern.

Let me know if you'd like more details or have questions!

### 5 Related Questions:
1. How can you determine if a relationship between two variables is linear or non-linear?
2. What methods can you use to find patterns in a number table?
3. How does piecewise notation work in mathematical functions?
4. Why do larger values of \( x \) seem to change the rule in this case?
5. Can you apply a different rule for values of \( x > 45 \)?

#### Tip: 
When faced with a non-linear table, always check if the relationship might involve multiple rules based on the value range of \( x \).

Guess the rule and complete the table given the pairs of x and y values.

Learn how to determine the algebraic rule to complete a table of x and y values. This problem involves understanding piecewise functions and pattern recognition to find the relationship between x and y. Suitable for students in Grades 6-8.

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