To find [tex]\( g^{-1}(19) \)[/tex] from the given table, we need to identify the value of [tex]\( x \)[/tex] for which [tex]\( g(x) = 19 \)[/tex].
Let's take a look at the table provided:
[tex]\[
\begin{array}{c|cccccc}
x & 4 & 7 & 12 & 14 & 16 & 19 \\
\hline
g(x) & 6 & 11 & 15 & 18 & 19 & 23
\end{array}
\][/tex]
We need to find the [tex]\( x \)[/tex] that corresponds to [tex]\( g(x) = 19 \)[/tex]:
- For [tex]\( x = 4 \)[/tex], [tex]\( g(4) = 6 \)[/tex]
- For [tex]\( x = 7 \)[/tex], [tex]\( g(7) = 11 \)[/tex]
- For [tex]\( x = 12 \)[/tex], [tex]\( g(12) = 15 \)[/tex]
- For [tex]\( x = 14 \)[/tex], [tex]\( g(14) = 18 \)[/tex]
- For [tex]\( x = 16 \)[/tex], [tex]\( g(16) = 19 \)[/tex]
- For [tex]\( x = 19 \)[/tex], [tex]\( g(19) = 23 \)[/tex]
From this examination, we see that [tex]\( g(16) = 19 \)[/tex]. Therefore, [tex]\( g^{-1}(19) = 16 \)[/tex].
So the correct answer is:
[tex]\( \boxed{g^{-1}(19) = 16} \)[/tex]
