To solve the problem, let's move step-by-step:
1. Understanding the Problem:
- We are given that the point [tex]$(8, 3)$[/tex] lies on the graph of [tex]\( g(x) = \log_2(x) \)[/tex]. This means [tex]\( \log_2(8) = 3 \)[/tex].
- We need to find the corresponding point that lies on the graph of [tex]\( f(x) = \log_2(x + 3) + 2 \)[/tex].
2. Starting with the Function [tex]\( g(x) \)[/tex]:
- By the given, [tex]\( \log_2(8) = 3 \)[/tex].
3. Finding [tex]\( y \)[/tex] in the Function [tex]\( f(x) \)[/tex] When [tex]\( x = 8 \)[/tex]:
- Now, we want to find the value of [tex]\( f(x) \)[/tex] at [tex]\( x = 8 \)[/tex].
[tex]\[
f(8) = \log_2(8 + 3) + 2
\][/tex]
- Calculate [tex]\( \log_2(11) \)[/tex]:
[tex]\[
\log_2(11) \approx 3.459431618637297
\][/tex]
- Plug this back into the equation for [tex]\( f(x) \)[/tex]:
[tex]\[
f(8) = \log_2(11) + 2 \approx 3.459431618637297 + 2 = 5.459431618637297
\][/tex]
- Thus, the point on the graph [tex]\( f(x) \)[/tex] corresponding to [tex]\( x = 8 \)[/tex] is [tex]\( (8, 5.459431618637297) \)[/tex].
4. Checking the Closest Point from the Given Options:
- The given points are: [tex]\( (5, 1) \)[/tex], [tex]\( (5, 5) \)[/tex], [tex]\( (11, 1) \)[/tex], and [tex]\( (11, 5) \)[/tex].
- We need to find the point among these which is closest to [tex]\( (8, 5.459431618637297) \)[/tex].
5. Calculate the Distance Using the Manhattan Distance:
- Distance to [tex]\( (5, 1) \)[/tex]:
[tex]\[
|8 - 5| + |5.459431618637297 - 1| = 3 + 4.459431618637297 = 7.459431618637297
\][/tex]
- Distance to [tex]\( (5, 5) \)[/tex]:
[tex]\[
|8 - 5| + |5.459431618637297 - 5| = 3 + 0.459431618637297 = 3.459431618637297
\][/tex]
- Distance to [tex]\( (11, 1) \)[/tex]:
[tex]\[
|8 - 11| + |5.459431618637297 - 1| = 3 + 4.459431618637297 = 7.459431618637297
\][/tex]
- Distance to [tex]\( (11, 5) \)[/tex]:
[tex]\[
|8 - 11| + |5.459431618637297 - 5| = 3 + 0.459431618637297 = 3.459431618637297
\][/tex]
6. Conclusion:
- The distances are calculated as follows:
- [tex]\( (5, 1) \)[/tex]: 7.459431618637297
- [tex]\( (5, 5) \)[/tex]: 3.459431618637297
- [tex]\( (11, 1) \)[/tex]: 7.459431618637297
- [tex]\( (11, 5) \)[/tex]: 3.459431618637297
- The closest points are [tex]\( (5, 5) \)[/tex] and [tex]\( (11, 5) \)[/tex], both equally distant.
Given the options and calculations, the closest corresponding point from the given options is [tex]\( (5, 5) \)[/tex].
