To find [tex]\((g \circ f)(3)\)[/tex], we need to follow a series of steps involving looking up values from the given table. Here's how you do it:
1. Find [tex]\( f(3) \)[/tex]:
Begin by looking into the table for the value of [tex]\( f \)[/tex] when [tex]\( x = 3 \)[/tex].
From the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
3 & 4 & 0 \\
\hline
\end{array}
\][/tex]
Therefore, [tex]\( f(3) = 4 \)[/tex].
2. Find [tex]\( g(f(3)) \)[/tex]:
Now, we need to find [tex]\( g \)[/tex] at the value we just found for [tex]\( f(3) \)[/tex]. So we need to determine [tex]\( g(4) \)[/tex].
From the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
4 & -3 & 2 \\
\hline
\end{array}
\][/tex]
Therefore, [tex]\( g(4) = 2 \)[/tex].
3. Combining these results:
Now we can combine the results to find [tex]\( (g \circ f)(3) \)[/tex]:
[tex]\[
(g \circ f)(3) = g(f(3)) = g(4) = 2
\][/tex]
Thus, the value of [tex]\( (g \circ f)(3) \)[/tex] is [tex]\( 2 \)[/tex].
