Janet has money invested in two different accounts. The function [tex]$ f(x) $[/tex] represents the amount of money in an account after [tex]$ x $[/tex] years of interest compounded monthly, while the function [tex]$ g(x) $[/tex] represents the amount of money in an account after [tex]$ x $[/tex] years of simple interest.

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$f(x)$ & 1,576.74 & 1,657.41 & 1,742.21 & 1,831.34 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$g(x)$ & 1,575 & 1,650 & 1,725 & 1,800 \\
\hline
\end{tabular}
\][/tex]

Complete the table to show the difference in the value of the accounts for each year.

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
Difference & 1.74 & 7.41 & 17.21 & 31.34 \\
\hline
\end{tabular}
\][/tex]

To find the difference in the value of Janet's accounts for each year, we need to calculate the difference between the amounts provided by the functions [tex]$ f(x) $[/tex] and [tex]$ g(x) $[/tex] for each year [tex]$ x $[/tex]. Here are the steps:

1. Identify the values of [tex]$ f(x) $[/tex] and [tex]$ g(x) $[/tex] for each year.
2. Calculate the difference [tex]$ f(x) - g(x) $[/tex] for each year.

Let's proceed step-by-step.

### Step 1: Identify the Values

For [tex]$ x = 1 $[/tex]:
- [tex]$ f(1) = 1576.74 $[/tex]
- [tex]$ g(1) = 1575 $[/tex]

For [tex]$ x = 2 $[/tex]:
- [tex]$ f(2) = 1657.41 $[/tex]
- [tex]$ g(2) = 1650 $[/tex]

For [tex]$ x = 3 $[/tex]:
- [tex]$ f(3) = 1742.21 $[/tex]
- [tex]$ g(3) = 1725 $[/tex]

For [tex]$ x = 4 $[/tex]:
- [tex]$ f(4) = 1831.34 $[/tex]
- [tex]$ g(4) = 1800 $[/tex]

### Step 2: Calculate the Differences

For [tex]$ x = 1 $[/tex]:
[tex]\[ f(1) - g(1) = 1576.74 - 1575 = 1.74 \][/tex]

For [tex]$ x = 2 $[/tex]:
[tex]\[ f(2) - g(2) = 1657.41 - 1650 = 7.41 \][/tex]

For [tex]$ x = 3 $[/tex]:
[tex]\[ f(3) - g(3) = 1742.21 - 1725 = 17.21 \][/tex]

For [tex]$ x = 4 $[/tex]:
[tex]\[ f(4) - g(4) = 1831.34 - 1800 = 31.34 \][/tex]

Now, complete the table with the calculated differences:

\begin{tabular}{|c|c|c|c|c|}
\hline [tex]$x$[/tex] & 1 & 2 & 3 & 4 \\
\hline Difference [tex]$ f(x) - g(x) $[/tex] & 1.74 & 7.41 & 17.21 & 31.34 \\
\hline
\end{tabular}

Thus, the differences in the value of the accounts for each year are as shown in the table above:
- For year 1: [tex]$1.74 - For year 2: $[/tex]7.41
- For year 3: [tex]$17.21 - For year 4: $[/tex]31.34