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!

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.