Question

The table below shows select values of the function ff, which is odd and whose domain is all real numbers. Find f(12).
x −12 −7 −4 9 13 14
f(x) 9 −12 −16 12 17 4

Answer 1

To find \( f(12) \) for the function \( f \) which is odd, we use the property of odd functions. An odd function satisfies the condition \( f(-x) = -f(x) \) for all \( x \).

Given the values in the table:

\[
\begin{align*}
f(-12) & = 9 \\
f(-7) & = -12 \\
f(-4) & = -16 \\
f(9) & = 12 \\
f(13) & = 17 \\
f(14) & = 4 \\
\end{align*}
\]

Using the property of odd functions, we can find values for \( f(-x) \):

\[
\begin{align*}
f(12) & = -f(-12) = -9 \quad \text{(since \( -12 \) corresponds to \( 12 \))} \\
f(7) & = -f(-7) = 12 \\
f(4) & = -f(-4) = 16 \\
f(-9) & = -f(9) = -12 \\
f(-13) & = -f(13) = -17 \\
f(-14) & = -f(14) = -4 \\
\end{align*}
\]

Now we have enough information to calculate \( f(12) \):
\[
f(12) = -f(-12) = -9
\]

Thus, the value of \( f(12) \) is

\[
\boxed{-9}.
\]