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Given that a function [tex]\( h \)[/tex] has a domain of [tex]\(-3 \leq x \leq 11\)[/tex] and a range of [tex]\(1 \leq h(x) \leq 25\)[/tex], and that [tex]\( h(8)=19 \)[/tex] and [tex]\( h(-2)=2 \)[/tex], select the statement that could be true for [tex]\( h \)[/tex].

A. [tex]\( h(8) = 21 \)[/tex]
B. [tex]\( h(-3) = -1 \)[/tex]
C. [tex]\( h(13) = 18 \)[/tex]
D. [tex]\( h(2) = 16 \)[/tex]


Sagot :

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]