Question

The table below shows select values of the function ff, which is odd and whose domain is all real numbers. Find minus, f, of, 3−f(3).
xx minus, 14−14 minus, 3−3 11 22 1212 1313
f, of, xf(x) minus, 3−3 minus, 8−8 33 minus, 5−5 minus, 3−3 minus, 2−2

Answer 1

To find \( -f(3 - f(3)) \), we can use the properties of odd functions and the values presented in the table you provided.

Since the function \( f \) is defined to be odd, we have the property that:

\[
f(-x) = -f(x)
\]

This means that for any \( x \), the function evaluated at \(-x\) will equal the negative of the function evaluated at \( x \).

From your table, it looks like you provided values of \( f(x) \) at certain points. However, it appears there might be a misunderstanding in how the values were presented. I will interpret the values as follows:

- \( f(-3) = -3 \)
- \( f(-2) = -2 \)
- \( f(-5) = -5 \)
- \( f(-8) = -8 \)
- \( f(-14) = -14 \)
- \( f(3) = 3 \)
- \( f(2) = 2 \)
- \( f(5) = 5 \)
- \( f(8) = 8 \)
- \( f(14) = 14 \)

Now, let's determine \( f(3) \) first (it looks like from the table that \( f(3) = 3 \)).

Next, we substitute this value into \( 3 - f(3) \):

\[
3 - f(3) = 3 - 3 = 0
\]

Now, we need to find \( -f(0) \).

Since the function is odd, we know that:

\[
f(0) = 0
\]

Thus, we have:

\[
-f(0) = -0 = 0
\]

So the final value of \( -f(3 - f(3)) \) is:

\[
\boxed{0}
\]