To determine which of the given values is in the domain of the function [tex]\( f(x) \)[/tex], we need to investigate the piecewise definition of [tex]\( f(x) \)[/tex]:
[tex]\[
f(x)=
\begin{cases}
2x + 5, & -6 < x \leq 0 \\
-2x + 3, & 0 < x \leq 4
\end{cases}
\][/tex]
### Step-by-Step Analysis:
1. Given Values:
- [tex]\( -7 \)[/tex]
- [tex]\( -6 \)[/tex]
- [tex]\( 4 \)[/tex]
- [tex]\( 5 \)[/tex]
2. Check if each value is in the domain of [tex]\( f(x) \)[/tex]:
- For [tex]\( -7 \)[/tex]:
- This value needs to satisfy either [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- Clearly, [tex]\( -7 \)[/tex] does not satisfy [tex]\( -6 < -7 \leq 0 \)[/tex], so [tex]\( -7 \)[/tex] is not in the domain.
- For [tex]\( -6 \)[/tex]:
- This value needs to satisfy either [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- Clearly, [tex]\( -6 \)[/tex] does not satisfy the strict inequality [tex]\( -6 < -6 \leq 0 \)[/tex], so [tex]\( -6 \)[/tex] is not in the domain.
- For [tex]\( 4 \)[/tex]:
- This value needs to satisfy either [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- [tex]\( 4 \)[/tex] satisfies the second interval [tex]\( 0 < 4 \leq 4 \)[/tex], so [tex]\( 4 \)[/tex] is in the domain.
- For [tex]\( 5 \)[/tex]:
- This value needs to satisfy either [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- Clearly, [tex]\( 5 \)[/tex] does not satisfy [tex]\( 0 < 5 \leq 4 \)[/tex], so [tex]\( 5 \)[/tex] is not in the domain.
### Conclusion:
The only value among the given options that falls within the specified domain of [tex]\( f(x) \)[/tex] is [tex]\( 4 \)[/tex]. Therefore, [tex]\( 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
