To determine which table represents the function \( g(x) \), given that \( g(x) \) is obtained by shifting \( f(x) \) 5 units down, let's analyze the changes step-by-step.
First, we look at the original values of \( f(x) \):
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
f(x) & 5 & 7 & 9 & 11 \\
\hline
\end{array}
\][/tex]
Shifting \( f(x) \) 5 units down means we subtract 5 from each value of \( f(x) \) to get \( g(x) \):
[tex]\[
g(x) = f(x) - 5
\][/tex]
Now, let's perform the subtraction for each \( f(x) \):
[tex]\[
\begin{align*}
g(1) &= f(1) - 5 = 5 - 5 = 0 \\
g(2) &= f(2) - 5 = 7 - 5 = 2 \\
g(3) &= f(3) - 5 = 9 - 5 = 4 \\
g(4) &= f(4) - 5 = 11 - 5 = 6 \\
\end{align*}
\][/tex]
So, the values of \( g(x) \) are:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
g(x) & 0 & 2 & 4 & 6 \\
\hline
\end{array}
\][/tex]
Comparing this with the provided choices:
A.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
g(x) & 10 & 12 & 14 & 16 \\
\hline
\end{array}
\][/tex]
B.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -4 & -3 & -2 & -1 \\
\hline
g(x) & 5 & 7 & 9 & 11 \\
\hline
\end{array}
\][/tex]
C.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
g(x) & 0 & 2 & 4 & 6 \\
\hline
\end{array}
\][/tex]
D.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -4 & -3 & -2 & -1 \\
\hline
g(x) & 0 & 2 & 4 & 6 \\
\hline
\end{array}
\][/tex]
The correct choice that matches our calculated values is:
Choice C.
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
g(x) & 0 & 2 & 4 & 6 \\
\hline
\end{array}
\][/tex]
