To solve for [tex]\( g(x) \)[/tex], we use the definitions given:
1. [tex]\( f(x) = 2x - 3 \)[/tex]
2. [tex]\( g(x) = f(3x) \)[/tex]
We need to calculate [tex]\( g(x) \)[/tex] for different values of [tex]\( x \)[/tex]:
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = f(3 \cdot 1) = f(3)
\][/tex]
Then, calculate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 2 \cdot 3 - 3 = 6 - 3 = 3
\][/tex]
Therefore:
[tex]\[
g(1) = 3
\][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = f(3 \cdot 2) = f(6)
\][/tex]
Then, calculate [tex]\( f(6) \)[/tex]:
[tex]\[
f(6) = 2 \cdot 6 - 3 = 12 - 3 = 9
\][/tex]
Therefore:
[tex]\[
g(2) = 9
\][/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = f(3 \cdot 3) = f(9)
\][/tex]
Then, calculate [tex]\( f(9) \)[/tex]:
[tex]\[
f(9) = 2 \cdot 9 - 3 = 18 - 3 = 15
\][/tex]
Therefore:
[tex]\[
g(3) = 15
\][/tex]
The correct table that represents [tex]\( g(x) \)[/tex] is:
[tex]\[
\begin{tabular}{|l|l|}
\hline
$x$ & $g(x)$ \\
\hline 1 & 3 \\
\hline 2 & 9 \\
\hline 3 & 15 \\
\hline
\end{tabular}
\][/tex]
