Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
[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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
