To determine which statement could be true for the function [tex]\( h \)[/tex] given its domain and range, we need to evaluate each option based on the constraints provided:
The domain is [tex]\( -3 \leq x \leq 11 \)[/tex].
The range is [tex]\( 1 \leq h(x) \leq 25 \)[/tex].
Additionally, specific values given are:
[tex]\( h(8) = 19 \)[/tex]
[tex]\( h(-2) = 2 \)[/tex]
Now, let's analyze each option individually:
### Option A: [tex]\( h(8) = 21 \)[/tex]
We know that the function [tex]\( h(x) \)[/tex] is defined for [tex]\( -3 \leq x \leq 11 \)[/tex], and the specific value [tex]\( h(8) \)[/tex] is given as [tex]\( 19 \)[/tex]. Option A states [tex]\( h(8) = 21 \)[/tex].
Although [tex]\( 21 \)[/tex] is within the range [tex]\( 1 \leq h(x) \leq 25 \)[/tex], this contradicts the provided value [tex]\( h(8) = 19 \)[/tex]. Therefore, Option A cannot be true.
### Option B: [tex]\( h(-3) = -1 \)[/tex]
Option B states [tex]\( h(-3) = -1 \)[/tex].
First, -3 is within the domain [tex]\( -3 \leq x \leq 11 \)[/tex]. However, the value [tex]\( -1 \)[/tex] is not within the range [tex]\( 1 \leq h(x) \leq 25 \)[/tex]. Therefore, Option B cannot be true.
### Option C: [tex]\( h(13) = 18 \)[/tex]
Option C states [tex]\( h(13) = 18 \)[/tex].
First, [tex]\( 13 \)[/tex] is not within the domain [tex]\( -3 \leq x \leq 11 \)[/tex]. Therefore, [tex]\( h(13) \)[/tex] is undefined since it's outside the domain. No matter the value provided for [tex]\( h(13) \)[/tex], since [tex]\( x = 13 \)[/tex] is not defined in the domain, it cannot be true.
### Option D: [tex]\( h(2) = 16 \)[/tex]
Option D states [tex]\( h(2) = 16 \)[/tex].
First, 2 is within the domain [tex]\( -3 \leq x \leq 11 \)[/tex]. Additionally, the value [tex]\( 16 \)[/tex] is within the range [tex]\( 1 \leq h(x) \leq 25 \)[/tex]. Nothing about these values contradicts the given information. Therefore, Option D could be true.
#### Conclusion:
The valid options are:
[tex]\( A \)[/tex]
[tex]\( C \)[/tex]
[tex]\( D \)[/tex]
None of the domain or range constraints, as well as the defined function values [tex]\( h(8) = 19 \)[/tex] and [tex]\( h(-2) = 2 \)[/tex], are violated by Option D. Therefore, the correct answer is [tex]\( D \)[/tex].
