To determine which values are in the domain of the function [tex]\( f(x) \)[/tex] as defined:
[tex]\[
f(x) = \begin{cases}
2x + 5, & -6 < x \leq 0 \\
-2x + 3, & 0 < x \leq 4
\end{cases}
\][/tex]
We'll examine each given value one by one to see if it falls within the appropriate intervals:
1. For [tex]\( x = -7 \)[/tex]:
- Check the conditions:
- [tex]\( -6 < -7 \leq 0 \)[/tex] is false because [tex]\(-7\)[/tex] is not greater than [tex]\(-6\)[/tex].
- [tex]\( 0 < -7 \leq 4 \)[/tex] is false because [tex]\(-7\)[/tex] is not greater than [tex]\(0\)[/tex].
- Conclusion: [tex]\( x = -7 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
2. For [tex]\( x = -6 \)[/tex]:
- Check the conditions:
- [tex]\( -6 < -6 \leq 0 \)[/tex] is false because [tex]\(-6\)[/tex] is not strictly greater than [tex]\(-6\)[/tex].
- [tex]\( 0 < -6 \leq 4 \)[/tex] is false because [tex]\(-6\)[/tex] is not greater than [tex]\(0\)[/tex].
- Conclusion: [tex]\( x = -6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
3. For [tex]\( x = 4 \)[/tex]:
- Check the conditions:
- [tex]\( -6 < 4 \leq 0 \)[/tex] is false because [tex]\(4\)[/tex] is not less than or equal to [tex]\(0\)[/tex].
- [tex]\( 0 < 4 \leq 4 \)[/tex] is true because [tex]\(4\)[/tex] is in the interval [tex]\(0 < x \leq 4\)[/tex].
- Conclusion: [tex]\( x = 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
4. For [tex]\( x = 5 \)[/tex]:
- Check the conditions:
- [tex]\( -6 < 5 \leq 0 \)[/tex] is false because [tex]\(5\)[/tex] is not less than or equal to [tex]\(0\)[/tex].
- [tex]\( 0 < 5 \leq 4 \)[/tex] is false because [tex]\(5\)[/tex] is not less than or equal to [tex]\(4\)[/tex].
- Conclusion: [tex]\( x = 5 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
Therefore, of the given values, only [tex]\( x = 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
