To determine which function rule models the function over the given domain, we will check each given rule against the provided points in the table.
The given points are:
[tex]\[
\begin{array}{|c|c|}
\hline x & f(x) \\
\hline -7 & -11 \\
\hline -1 & 1 \\
\hline 3 & 9 \\
\hline 4 & 11 \\
\hline 7 & 17 \\
\hline
\end{array}
\][/tex]
Let's test each function rule one by one:
1. Rule: [tex]\(f(x) = 3x + 10\)[/tex]
[tex]\[
\begin{align*}
f(-7) &= 3(-7) + 10 = -21 + 10 = -11 \quad (\text{matches}) \\
f(-1) &= 3(-1) + 10 = -3 + 10 = 7 \quad (\text{does not match}) \\
f(3) &= 3(3) + 10 = 9 + 10 = 19 \quad (\text{does not match}) \\
f(4) &= 3(4) + 10 = 12 + 10 = 22 \quad (\text{does not match}) \\
f(7) &= 3(7) + 10 = 21 + 10 = 31 \quad (\text{does not match})
\end{align*}
\][/tex]
The rule [tex]\(f(x) = 3x + 10\)[/tex] does not fit all points.
2. Rule: [tex]\(f(x) = 2x + 3\)[/tex]
[tex]\[
\begin{align*}
f(-7) &= 2(-7) + 3 = -14 + 3 = -11 \quad (\text{matches}) \\
f(-1) &= 2(-1) + 3 = -2 + 3 = 1 \quad (\text{matches}) \\
f(3) &= 2(3) + 3 = 6 + 3 = 9 \quad (\text{matches}) \\
f(4) &= 2(4) + 3 = 8 + 3 = 11 \quad (\text{matches}) \\
f(7) &= 2(7) + 3 = 14 + 3 = 17 \quad (\text{matches})
\end{align*}
\][/tex]
The rule [tex]\(f(x) = 2x + 3\)[/tex] fits all points.
3. Rule: [tex]\(f(x) = 4x + 5\)[/tex]
[tex]\[
\begin{align*}
f(-7) &= 4(-7) + 5 = -28 + 5 = -23 \quad (\text{does not match}) \\
f(-1) &= 4(-1) + 5 = -4 + 5 = 1 \quad (\text{matches}) \\
f(3) &= 4(3) + 5 = 12 + 5 = 17 \quad (\text{does not match}) \\
f(4) &= 4(4) + 5 = 16 + 5 = 21 \quad (\text{does not match}) \\
f(7) &= 4(7) + 5 = 28 + 5 = 33 \quad (\text{does not match})
\end{align*}
\][/tex]
The rule [tex]\(f(x) = 4x + 5\)[/tex] does not fit all points.
4. Rule: [tex]\(f(x) = 3x - 10\)[/tex]
[tex]\[
\begin{align*}
f(-7) &= 3(-7) - 10 = -21 - 10 = -31 \quad (\text{does not match}) \\
f(-1) &= 3(-1) - 10 = -3 - 10 = -13 \quad (\text{does not match}) \\
f(3) &= 3(3) - 10 = 9 - 10 = -1 \quad (\text{does not match}) \\
f(4) &= 3(4) - 10 = 12 - 10 = 2 \quad (\text{does not match}) \\
f(7) &= 3(7) - 10 = 21 - 10 = 11 \quad (\text{does not match})
\end{align*}
\][/tex]
The rule [tex]\(f(x) = 3x - 10\)[/tex] does not fit all points.
Based on this detailed step-by-step evaluation, the function rule that models the function over the domain specified in the table is:
[tex]\[
f(x) = 2x + 3
\][/tex]
