Let's analyze the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at different points [tex]\( x \)[/tex]. We need to determine the corresponding values for [tex]\( h(x) = f(x) + g(x) \)[/tex].
The given table is:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & $f(x)$ & $g(x)$ & $h(x) = f(x) + g(x)$ \\
\hline
1 & $3 \frac{1}{2}$ & 1 & \\
\hline
2 & 4 & $\frac{1}{4}$ & \\
\hline
3 & $4 \frac{1}{2}$ & $\frac{1}{9}$ & \\
\hline
4 & 5 & $\frac{1}{16}$ & \\
\hline
\end{tabular}
\][/tex]
We need to calculate the values of [tex]\( h(x) \)[/tex] for each [tex]\( x \)[/tex].
For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3 \frac{1}{2} = 3 + \frac{1}{2} = 3.5 \][/tex]
[tex]\[ g(1) = 1 \][/tex]
[tex]\[ h(1) = f(1) + g(1) = 3.5 + 1 = 4.5 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 4 \][/tex]
[tex]\[ g(2) = \frac{1}{4} = 0.25 \][/tex]
[tex]\[ h(2) = f(2) + g(2) = 4 + 0.25 = 4.25 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 4 \frac{1}{2} = 4 + \frac{1}{2} = 4.5 \][/tex]
[tex]\[ g(3) = \frac{1}{9} \approx 0.1111111111111111 \][/tex]
[tex]\[ h(3) = f(3) + g(3) \approx 4.5 + 0.1111111111111111 \approx 4.611111111111111 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 5 \][/tex]
[tex]\[ g(4) = \frac{1}{16} = 0.0625 \][/tex]
[tex]\[ h(4) = f(4) + g(4) = 5 + 0.0625 = 5.0625 \][/tex]
So, the completed table should be:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & $f(x)$ & $g(x)$ & $h(x) = f(x) + g(x)$ \\
\hline
1 & $3 \frac{1}{2}$ & 1 & 4.5 \\
\hline
2 & 4 & $\frac{1}{4}$ & 4.25 \\
\hline
3 & $4 \frac{1}{2}$ & $\frac{1}{9}$ & 4.611111111111111 \\
\hline
4 & 5 & $\frac{1}{16}$ & 5.0625 \\
\hline
\end{tabular}
\][/tex]
Therefore, the values that complete the table for [tex]\( h(x) \)[/tex] are:
[tex]\[
\begin{tabular}{|c|}
\hline
$h(x) = f(x) + g(x)$ \\
\hline
4.5 \\
\hline
4.25 \\
\hline
4.611111111111111 \\
\hline
5.0625 \\
\hline
\end{tabular}
\][/tex]
