To solve this problem, we need to examine each [tex]\( x \)[/tex]-value and determine whether the output of the floor function [tex]\( g(x) = \lfloor x \rfloor \)[/tex] is equal to the output of the ceiling function [tex]\( h(x) = \lceil x \rceil \)[/tex].
Here's a step-by-step evaluation for each [tex]\( x \)[/tex]-value:
1. For [tex]\( x = -8 \)[/tex]:
- The floor function [tex]\( g(x) = \lfloor -8 \rfloor \)[/tex] is [tex]\(-8\)[/tex].
- The ceiling function [tex]\( h(x) = \lceil -8 \rceil \)[/tex] is also [tex]\(-8\)[/tex].
- Since [tex]\( \lfloor -8 \rfloor = \lceil -8 \rceil = -8 \)[/tex], the values are equal.
2. For [tex]\( x = -5.2 \)[/tex]:
- The floor function [tex]\( g(x) = \lfloor -5.2 \rfloor \)[/tex] is [tex]\(-6\)[/tex].
- The ceiling function [tex]\( h(x) = \lceil -5.2 \rceil \)[/tex] is [tex]\(-5\)[/tex].
- Since [tex]\( \lfloor -5.2 \rfloor \neq \lceil -5.2 \rceil \)[/tex], the values are not equal.
3. For [tex]\( x = -1.7 \)[/tex]:
- The floor function [tex]\( g(x) = \lfloor -1.7 \rfloor \)[/tex] is [tex]\(-2\)[/tex].
- The ceiling function [tex]\( h(x) = \lceil -1.7 \rceil \)[/tex] is [tex]\(-1\)[/tex].
- Since [tex]\( \lfloor -1.7 \rfloor \neq \lceil -1.7 \rceil \)[/tex], the values are not equal.
4. For [tex]\( x = 0 \)[/tex]:
- The floor function [tex]\( g(x) = \lfloor 0 \rfloor \)[/tex] is [tex]\(0\)[/tex].
- The ceiling function [tex]\( h(x) = \lceil 0 \rceil \)[/tex] is also [tex]\(0\)[/tex].
- Since [tex]\( \lfloor 0 \rfloor = \lceil 0 \rceil = 0 \)[/tex], the values are equal.
5. For [tex]\( x = 2.4 \)[/tex]:
- The floor function [tex]\( g(x) = \lfloor 2.4 \rfloor \)[/tex] is [tex]\(2\)[/tex].
- The ceiling function [tex]\( h(x) = \lceil 2.4 \rceil \)[/tex] is [tex]\(3\)[/tex].
- Since [tex]\( \lfloor 2.4 \rfloor \neq \lceil 2.4 \rceil \)[/tex], the values are not equal.
Based on this evaluation, the [tex]\( x \)[/tex]-values where the floor and ceiling functions are equal are:
[tex]\[ -8 \][/tex]
[tex]\[ 0 \][/tex]
Thus, the correct [tex]\( x \)[/tex]-values are [tex]\(-8\)[/tex] and [tex]\(0\)[/tex].
