Let's carefully analyze each given statement and compare them to the known values and constraints of the function [tex]\( g \)[/tex].
We know the following:
- Domain of [tex]\( g \)[/tex]: [tex]\(-20 \leq x \leq 5\)[/tex]
- Range of [tex]\( g \)[/tex]: [tex]\(-5 \leq g(x) \leq 45\)[/tex]
- Specific values: [tex]\( g(0) = -2 \)[/tex] and [tex]\( g(-9) = 6 \)[/tex]
Let's look at each statement:
### Statement A: [tex]\( g(0) = 2 \)[/tex]
We are given that [tex]\( g(0) = -2 \)[/tex]. Therefore, it contradicts the statement [tex]\( g(0) = 2 \)[/tex]. This statement is False.
### Statement B: [tex]\( g(7) = -1 \)[/tex]
The domain of [tex]\( g \)[/tex] is [tex]\(-20 \leq x \leq 5\)[/tex]. The value [tex]\( x = 7 \)[/tex] falls outside this domain; hence, it is not possible to evaluate [tex]\( g(7) \)[/tex] within the given domain. This statement is also False.
### Statement C: [tex]\( g(-4) = -11 \)[/tex]
The range of [tex]\( g \)[/tex] is [tex]\( -5 \leq g(x) \leq 45 \)[/tex]. The value [tex]\( g(-4) = -11 \)[/tex] falls outside of this range. Hence, this statement cannot be true. This statement is False.
### Statement D: [tex]\( g(-13) = 20 \)[/tex]
The domain of [tex]\( g \)[/tex] is [tex]\(-20 \leq x \leq 5\)[/tex]. The value [tex]\( x = -13 \)[/tex] falls within this range. Also, [tex]\( g(-13) = 20 \)[/tex] falls within the range of [tex]\( -5 \leq g(x) \leq 45 \)[/tex]. There is no given information that contradicts this statement. Thus, this statement could be True.
Thus, based on all the given information and constraints, the statement that could be true is:
[tex]\[ \boxed{D} \][/tex]
