Given the table of values for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
3 & 2 \\
\hline
5 & 3 \\
\hline
7 & 4 \\
\hline
9 & 5 \\
\hline
\end{array}
\][/tex]
We need to identify which of the given functions could be the correct [tex]\( f(x) \)[/tex]:
1. [tex]\( f(x) = \frac{1}{2} x - \frac{1}{2} \)[/tex]
2. [tex]\( f(x) = \frac{1}{2} x + \frac{1}{2} \)[/tex]
3. [tex]\( f(x) = \frac{2}{3} x \)[/tex]
4. [tex]\( f(x) = \frac{3}{2} x - \frac{5}{2} \)[/tex]
To determine the correct function, we will test each function with the given [tex]\( x \)[/tex] values (3, 5, 7, 9) and see which one produces the corresponding [tex]\( f(x) \)[/tex] values (2, 3, 4, 5).
### Testing Each Function
1. [tex]\( f(x) = \frac{1}{2} x - \frac{1}{2} \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{1}{2}(3) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1 \][/tex]
This does not match the given value [tex]\( f(3) = 2 \)[/tex].
- Since [tex]\( f(3) \neq 2 \)[/tex], [tex]\( f(x) = \frac{1}{2} x - \frac{1}{2} \)[/tex] is not the correct function.
2. [tex]\( f(x) = \frac{1}{2} x + \frac{1}{2} \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{1}{2}(3) + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = 2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{1}{2}(5) + \frac{1}{2} = \frac{5}{2} + \frac{1}{2} = 3 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = \frac{1}{2}(7) + \frac{1}{2} = \frac{7}{2} + \frac{1}{2} = 4 \][/tex]
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ f(9) = \frac{1}{2}(9) + \frac{1}{2} = \frac{9}{2} + \frac{1}{2} = 5 \][/tex]
Since these values match the given [tex]\( f(x) \)[/tex] values exactly, [tex]\( f(x) = \frac{1}{2} x + \frac{1}{2} \)[/tex] is a possible function.
3. [tex]\( f(x) = \frac{2}{3} x \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{2}{3}(3) = 2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{2}{3}(5) \approx 3.33 \][/tex]
Since [tex]\( f(5) \neq 3 \)[/tex], [tex]\( f(x) = \frac{2}{3} x \)[/tex] is not the correct function.
4. [tex]\( f(x) = \frac{3}{2} x - \frac{5}{2} \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = \frac{3}{2}(3) - \frac{5}{2} = \frac{9}{2} - \frac{5}{2} = 2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{3}{2}(5) - \frac{5}{2} = \frac{15}{2} - \frac{5}{2} = 5 \][/tex]
Since [tex]\( f(5) \neq 3 \)[/tex], [tex]\( f(x) = \frac{3}{2} x - \frac{5}{2} \)[/tex] is not the correct function.
### Conclusion
The only function that satisfies all the given values is:
[tex]\[ f(x) = \frac{1}{2} x + \frac{1}{2} \][/tex]
Thus, the correct function is:
[tex]\[ f(x) = \frac{1}{2} x + \frac{1}{2} \][/tex]
