Let's determine the correct function rule that matches the given (x, f(x)) pairs. Given the table:
[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]
We need to check each proposed function rule to see which one correctly calculates [tex]\(f(x)\)[/tex] for all given x values.
### Checking [tex]\(f(x) = 3x + 10\)[/tex]
- When [tex]\(x = -7\)[/tex]:
[tex]\[
f(-7) = 3(-7) + 10 = -21 + 10 = -11
\][/tex]
Matches the table value.
- When [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = 3(-1) + 10 = -3 + 10 = 7
\][/tex]
Does not match the table value (table value is 1).
Since there is no match for [tex]\(x = -1\)[/tex], [tex]\(f(x) = 3x + 10\)[/tex] is not the correct function rule.
### Checking [tex]\(f(x) = 2x + 3\)[/tex]
- When [tex]\(x = -7\)[/tex]:
[tex]\[
f(-7) = 2(-7) + 3 = -14 + 3 = -11
\][/tex]
Matches the table value.
- When [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = 2(-1) + 3 = -2 + 3 = 1
\][/tex]
Matches the table value.
- When [tex]\(x = 3\)[/tex]:
[tex]\[
f(3) = 2(3) + 3 = 6 + 3 = 9
\][/tex]
Matches the table value.
- When [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = 2(4) + 3 = 8 + 3 = 11
\][/tex]
Matches the table value.
- When [tex]\(x = 7\)[/tex]:
[tex]\[
f(7) = 2(7) + 3 = 14 + 3 = 17
\][/tex]
Matches the table value.
Since all values match the table, [tex]\(f(x) = 2x + 3\)[/tex] is the correct function rule.
### Checking [tex]\(f(x) = 4x + 5\)[/tex]
- When [tex]\(x = -7\)[/tex]:
[tex]\[
f(-7) = 4(-7) + 5 = -28 + 5 = -23
\][/tex]
Does not match the table value (table value is -11).
Since there is no match for [tex]\(x = -7\)[/tex], [tex]\(f(x) = 4x + 5\)[/tex] is not the correct function rule.
### Checking [tex]\(f(x) = 3x - 10\)[/tex]
- When [tex]\(x = -7\)[/tex]:
[tex]\[
f(-7) = 3(-7) - 10 = -21 - 10 = -31
\][/tex]
Does not match the table value (table value is -11).
Since there is no match for [tex]\(x = -7\)[/tex], [tex]\(f(x) = 3x - 10\)[/tex] is not the correct function rule.
After checking all the function rules, we determine that the correct function that models the data given in the table is:
[tex]\[
f(x) = 2x + 3
\][/tex]
