https://file.aiquickdraw.com/mathbot/file/17266962589707ifndznx.jpg

We are given that \( g(x) = f(-x) \), and we need to use this information to complete the table.

The function \( g(x) \) is essentially the reflection of \( f(x) \) across the \( y \)-axis. So, to find \( g(x) \), we look at \( f(-x) \). Here's how we'll proceed:

### Steps:
1. Take the given value of \( x \) for \( g(x) \).
2. Find the corresponding value of \( -x \).
3. Look up \( f(-x) \) in the table to get the value of \( g(x) \).

### Table from image:
- \( f(x) \) values for corresponding \( x \):
  - \( f(15) = -4 \)
  - \( f(-8) = -9 \)
  - \( f(-10) = 8 \)
  - \( f(11) = -6 \)

### Solution:
Now, we'll use \( g(x) = f(-x) \) to fill the table.

For each value of \( x \) in the lower table, we'll use \( g(x) = f(-x) \):
- When \( x = 15 \), \( g(15) = f(-15) \) (but we don't have \( f(-15) \) in the table, so this value remains unknown).
- When \( x = -8 \), \( g(-8) = f(8) \) (again, \( f(8) \) is unknown from the given table).
- When \( x = -10 \), \( g(-10) = f(10) \) (we don't have \( f(10) \)).
- When \( x = 11 \), \( g(11) = f(-11) \) (similarly, \( f(-11) \) is not provided).

It seems there are missing \( f(x) \) values for the reflected \( x \)-values needed to complete the lower table. Therefore, without the values of \( f(x) \) for these negative inputs, I can't fully complete the table. 

Would you like to attempt to infer the missing values, or should I help with a different method? Let me know!

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Here are 5 related questions to expand on this:
1. What is the difference between even and odd functions in terms of their symmetry?
2. How can we graphically interpret the function \( g(x) = f(-x) \)?
3. What happens to a function when it is reflected across the y-axis?
4. How would you verify if a function is even or odd using its algebraic expression?
5. Can a function be both even and odd? Why or why not?

**Tip**: Always check for patterns or symmetries in functions when transforming them, as it often simplifies problem-solving.

Complete the table given that g(x) = f(-x), using the provided f(x) values.

This problem involves completing a table using the reflection principle, where g(x) = f(-x). It tests understanding of function transformations and reflection across the y-axis. Suitable for high school students in grades 9-12.

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