To determine which function rule models the given data, we need to check each function against the data points provided in the table. Let's analyze each function:
### Step-by-Step Analysis:
Given Data Points:
[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]
Function Rules:
1. [tex]\( f(x) = 3x + 10 \)[/tex]
2. [tex]\( f(x) = 2x + 3 \)[/tex]
3. [tex]\( f(x) = 4x + 5 \)[/tex]
4. [tex]\( f(x) = 3x - 10 \)[/tex]
### Check each function with the data points:
1. Function: [tex]\( f(x) = 3x + 10 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( 3(-7) + 10 = -21 + 10 = -11 \)[/tex] ✓
- For [tex]\( x = -1 \)[/tex]: [tex]\( 3(-1) + 10 = -3 + 10 = 7 \)[/tex] ✗
- For [tex]\( x = 3 \)[/tex]: [tex]\( 3(3) + 10 = 9 + 10 = 19 \)[/tex] ✗
- For [tex]\( x = 4 \)[/tex]: [tex]\( 3(4) + 10 = 12 + 10 = 22 \)[/tex] ✗
- For [tex]\( x = 7 \)[/tex]: [tex]\( 3(7) + 10 = 21 + 10 = 31 \)[/tex] ✗
2. Function: [tex]\( f(x) = 2x + 3 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( 2(-7) + 3 = -14 + 3 = -11 \)[/tex] ✓
- For [tex]\( x = -1 \)[/tex]: [tex]\( 2(-1) + 3 = -2 + 3 = 1 \)[/tex] ✓
- For [tex]\( x = 3 \)[/tex]: [tex]\( 2(3) + 3 = 6 + 3 = 9 \)[/tex] ✓
- For [tex]\( x = 4 \)[/tex]: [tex]\( 2(4) + 3 = 8 + 3 = 11 \)[/tex] ✓
- For [tex]\( x = 7 \)[/tex]: [tex]\( 2(7) + 3 = 14 + 3 = 17 \)[/tex] ✓
3. Function: [tex]\( f(x) = 4x + 5 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( 4(-7) + 5 = -28 + 5 = -23 \)[/tex] ✗
- For [tex]\( x = -1 \)[/tex]: [tex]\( 4(-1) + 5 = -4 + 5 = 1 \)[/tex] ✓
- For [tex]\( x = 3 \)[/tex]: [tex]\( 4(3) + 5 = 12 + 5 = 17 \)[/tex] ✗
- For [tex]\( x = 4 \)[/tex]: [tex]\( 4(4) + 5 = 16 + 5 = 21 \)[/tex] ✗
- For [tex]\( x = 7 \)[/tex]: [tex]\( 4(7) + 5 = 28 + 5 = 33 \)[/tex] ✗
4. Function: [tex]\( f(x) = 3x - 10 \)[/tex]
- For [tex]\( x = -7 \)[/tex]: [tex]\( 3(-7) - 10 = -21 - 10 = -31 \)[/tex] ✗
- For [tex]\( x = -1 \)[/tex]: [tex]\( 3(-1) - 10 = -3 - 10 = -13 \)[/tex] ✗
- For [tex]\( x = 3 \)[/tex]: [tex]\( 3(3) - 10 = 9 - 10 = -1 \)[/tex] ✗
- For [tex]\( x = 4 \)[/tex]: [tex]\( 3(4) - 10 = 12 - 10 = 2 \)[/tex] ✗
- For [tex]\( x = 7 \)[/tex]: [tex]\( 3(7) - 10 = 21 - 10 = 11 \)[/tex] ✗
### Conclusion:
By checking each function against the given data points, we find that the function [tex]\( f(x) = 2x + 3 \)[/tex] correctly models the function over the domain specified in the table.
Therefore, the function rule that models the given data points is:
[tex]\[ f(x) = 2x + 3 \][/tex]
