To find which function [tex]\( f(x) \)[/tex] correctly represents the data given in the table, we can evaluate each function using the values of miles [tex]\( x \)[/tex] and see if the calculated costs match the provided total costs.
The table is given as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Miles},\, x & \text{Total Cost},\, f(x) \\
\hline
0 & \$10 \\
\hline
4 & \$12 \\
\hline
8 & \$14 \\
\hline
12 & \$16 \\
\hline
\end{array}
\][/tex]
Let's check each proposed function:
### Option A: [tex]\( f(x) = 0.5x + 10 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0.5 \times 0 + 10 = 10
\][/tex]
Matches the table: [tex]\( \$10 \)[/tex].
2. For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 0.5 \times 4 + 10 = 2 + 10 = 12
\][/tex]
Matches the table: [tex]\( \$. 12 \)[/tex].
3. For [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = 0.5 \times 8 + 10 = 4 + 10 = 14
\][/tex]
Matches the table: [tex]\( \$. 14 \)[/tex].
4. For [tex]\( x = 12 \)[/tex]:
[tex]\[
f(12) = 0.5 \times 12 + 10 = 6 + 10 = 16
\][/tex]
Matches the table: [tex]\( \$. 16 \)[/tex].
Since all the evaluations match the values in the table, the correct function is:
[tex]\[
\boxed{f(x) = 0.5x + 10}
\][/tex]
