To determine which value is in the domain of the function [tex]\( f(x) \)[/tex], we need to examine each given value and see if it falls within the intervals defined by the piecewise function.
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) =
\begin{cases}
2x + 5 & \text{if } -6 < x \leq 0 \\
-2x + 3 & \text{if } 0 < x \leq 4
\end{cases}
\][/tex]
Let's analyze each value one by one:
1. Value: -7
- We check if [tex]\( -7 \)[/tex] falls within any of the intervals [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- [tex]\( -7 \)[/tex] is less than [tex]\( -6 \)[/tex], so it does not satisfy [tex]\( -6 < x \leq 0 \)[/tex].
- [tex]\( -7 \)[/tex] also does not satisfy [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( -7 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
2. Value: -6
- We check if [tex]\( -6 \)[/tex] falls within any of the intervals [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- [tex]\( -6 \)[/tex] equals the lower bound of the interval [tex]\( -6 < x \leq 0 \)[/tex], but this interval is exclusive of [tex]\( -6 \)[/tex].
- [tex]\( -6 \)[/tex] does not satisfy [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( -6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
3. Value: 4
- We check if [tex]\( 4 \)[/tex] falls within any of the intervals [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- [tex]\( 4 \)[/tex] does not satisfy [tex]\( -6 < x \leq 0 \)[/tex].
- [tex]\( 4 \)[/tex] does satisfy [tex]\( 0 < x \leq 4 \)[/tex], as it is an inclusive interval.
- Therefore, [tex]\( 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
4. Value: 5
- We check if [tex]\( 5 \)[/tex] falls within any of the intervals [tex]\( -6 < x \leq 0 \)[/tex] or [tex]\( 0 < x \leq 4 \)[/tex].
- [tex]\( 5 \)[/tex] does not satisfy [tex]\( -6 < x \leq 0 \)[/tex].
- [tex]\( 5 \)[/tex] also does not satisfy [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( 5 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
Thus, after analyzing each value, the value [tex]\( 4 \)[/tex] is in the domain of the function [tex]\( f(x) \)[/tex].
