Answer :
To determine which statement could be true for the function [tex]\( h \)[/tex], we need to carefully check each option against the given conditions for the domain and range.
The domain of the function [tex]\( h \)[/tex] is:
[tex]\[ -3 \leq x \leq 11 \][/tex]
The range of the function [tex]\( h \)[/tex] is:
[tex]\[ 1 \leq h(x) \leq 25 \][/tex]
Additionally, we know:
[tex]\[ h(8) = 19 \][/tex]
[tex]\[ h(-2) = 2 \][/tex]
Now, we will evaluate each given statement:
Option A: [tex]\( h(8) = 21 \)[/tex]
We see that [tex]\( h(8) = 19 \)[/tex] is already given. Therefore, [tex]\( h(8) = 21 \)[/tex] is false since it contradicts the given value.
Option B: [tex]\( h(-3) = -1 \)[/tex]
According to the range of [tex]\( h \)[/tex]:
[tex]\[ 1 \leq h(x) \leq 25 \][/tex]
If [tex]\( h(-3) = -1 \)[/tex], it falls outside the range since -1 is less than 1. Hence, [tex]\( h(-3) = -1 \)[/tex] is false.
Option C: [tex]\( h(13) = 18 \)[/tex]
From the domain of [tex]\( h \)[/tex]:
[tex]\[ -3 \leq x \leq 11 \][/tex]
The value [tex]\( x = 13 \)[/tex] is outside this domain. Hence, [tex]\( h(13) = 18 \)[/tex] is false because [tex]\( x \)[/tex] must be within the domain of [tex]\( -3 \)[/tex] to [tex]\( 11 \)[/tex].
Option D: [tex]\( h(2) = 16 \)[/tex]
We need to check if this [tex]\( x \)[/tex]-value is within the domain and the associated [tex]\( h(x) \)[/tex]-value is within the range:
- The value [tex]\( x = 2 \)[/tex] lies within the domain [tex]\( -3 \leq x \leq 11 \)[/tex].
- The value [tex]\( h(2) = 16 \)[/tex] lies within the range [tex]\( 1 \leq h(x) \leq 25 \)[/tex].
Since both conditions are satisfied, this option does not contradict the information given and thus could be true.
Therefore, the correct option is:
[tex]\[ \boxed{D} \][/tex]
The domain of the function [tex]\( h \)[/tex] is:
[tex]\[ -3 \leq x \leq 11 \][/tex]
The range of the function [tex]\( h \)[/tex] is:
[tex]\[ 1 \leq h(x) \leq 25 \][/tex]
Additionally, we know:
[tex]\[ h(8) = 19 \][/tex]
[tex]\[ h(-2) = 2 \][/tex]
Now, we will evaluate each given statement:
Option A: [tex]\( h(8) = 21 \)[/tex]
We see that [tex]\( h(8) = 19 \)[/tex] is already given. Therefore, [tex]\( h(8) = 21 \)[/tex] is false since it contradicts the given value.
Option B: [tex]\( h(-3) = -1 \)[/tex]
According to the range of [tex]\( h \)[/tex]:
[tex]\[ 1 \leq h(x) \leq 25 \][/tex]
If [tex]\( h(-3) = -1 \)[/tex], it falls outside the range since -1 is less than 1. Hence, [tex]\( h(-3) = -1 \)[/tex] is false.
Option C: [tex]\( h(13) = 18 \)[/tex]
From the domain of [tex]\( h \)[/tex]:
[tex]\[ -3 \leq x \leq 11 \][/tex]
The value [tex]\( x = 13 \)[/tex] is outside this domain. Hence, [tex]\( h(13) = 18 \)[/tex] is false because [tex]\( x \)[/tex] must be within the domain of [tex]\( -3 \)[/tex] to [tex]\( 11 \)[/tex].
Option D: [tex]\( h(2) = 16 \)[/tex]
We need to check if this [tex]\( x \)[/tex]-value is within the domain and the associated [tex]\( h(x) \)[/tex]-value is within the range:
- The value [tex]\( x = 2 \)[/tex] lies within the domain [tex]\( -3 \leq x \leq 11 \)[/tex].
- The value [tex]\( h(2) = 16 \)[/tex] lies within the range [tex]\( 1 \leq h(x) \leq 25 \)[/tex].
Since both conditions are satisfied, this option does not contradict the information given and thus could be true.
Therefore, the correct option is:
[tex]\[ \boxed{D} \][/tex]
