Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.