Question

The table below shows select values of the function ff, which is odd and whose domain is all real numbers. Find f, of, 11f(11).
x −11 −10 −9 −7 −5 11
f(x) 9 −18 2 9 11−11

Answer 1

To reconstruct the values of the odd function \( f \) using the provided table, we need to use the property that \( f(-x) = -f(x) \) for odd functions.

Given the provided values:

- \( f(-11) = 9 \)
- \( f(-10) = -18 \)
- \( f(-9) = 2 \)
- \( f(-7) = 9 \)
- \( f(-5) = -11 \)
- \( f(11) = ? \)

Let's convert the given values where \( x \) is negative to find their corresponding positive \( x \) values:

1. From \( f(-11) = 9 \), we have:
\[
f(11) = -f(-11) = -9
\]

2. From \( f(-10) = -18 \), we have:
\[
f(10) = -f(-10) = 18
\]

3. From \( f(-9) = 2 \), we have:
\[
f(9) = -f(-9) = -2
\]

4. From \( f(-7) = 9 \), we have:
\[
f(7) = -f(-7) = -9
\]

5. From \( f(-5) = -11 \), we have:
\[
f(5) = -f(-5) = 11
\]

Now we can summarize the values of the function \( f(x) \):
- \( f(5) = 11 \)
- \( f(7) = -9 \)
- \( f(9) = -2 \)
- \( f(10) = 18 \)
- \( f(11) = -9 \)

Finally, we can compute \( f(f(11)) \):
Since \( f(11) = -9 \), we need to find \( f(-9) \):
From the table, \( f(-9) = 2 \), therefore:
\[
f(f(11)) = f(-9) = 2
\]

Thus, the final results are:
- \( f(11) = -9 \)
- \( f(f(11)) = 2 \)