Answer :
To determine which value is in the domain of the function [tex]\( f(x) \)[/tex], defined as:
[tex]\[ f(x) = \begin{cases} 2x + 5, & -6 < x \leq 0 \\ -2x + 3, & 0 < x \leq 4 \end{cases} \][/tex]
we need to evaluate each of the provided values and see if they fit within either interval of the piecewise function.
### Step-by-Step Analysis:
1. Consider [tex]\( x = -7 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( -7 \)[/tex] is not greater than [tex]\(-6\)[/tex]. Thus, [tex]\( -7 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( -7 \)[/tex] is not greater than 0. Thus, [tex]\( -7 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = -7 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
2. Consider [tex]\( x = -6 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( -6 \)[/tex] is not strictly greater than [tex]\(-6\)[/tex]. Since [tex]\( x \)[/tex] must be greater than [tex]\(-6\)[/tex], [tex]\( -6 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( -6 \)[/tex] is not greater than 0. Thus, [tex]\( -6 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = -6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
3. Consider [tex]\( x = 4 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( 4 \)[/tex] is greater than 0. Thus, [tex]\( 4 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( 4 \)[/tex] is less than or equal to [tex]\(4\)[/tex]. Thus, [tex]\( 4 \)[/tex] is included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
4. Consider [tex]\( x = 5 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( 5 \)[/tex] is greater than 0. Thus, [tex]\( 5 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( 5 \)[/tex] is greater than [tex]\( 4 \)[/tex]. Thus, [tex]\( 5 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = 5 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
### Conclusion:
After checking each value, we find that the value [tex]\( x = 4 \)[/tex] is the only value that is in the domain 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]
we need to evaluate each of the provided values and see if they fit within either interval of the piecewise function.
### Step-by-Step Analysis:
1. Consider [tex]\( x = -7 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( -7 \)[/tex] is not greater than [tex]\(-6\)[/tex]. Thus, [tex]\( -7 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( -7 \)[/tex] is not greater than 0. Thus, [tex]\( -7 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = -7 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
2. Consider [tex]\( x = -6 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( -6 \)[/tex] is not strictly greater than [tex]\(-6\)[/tex]. Since [tex]\( x \)[/tex] must be greater than [tex]\(-6\)[/tex], [tex]\( -6 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( -6 \)[/tex] is not greater than 0. Thus, [tex]\( -6 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = -6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
3. Consider [tex]\( x = 4 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( 4 \)[/tex] is greater than 0. Thus, [tex]\( 4 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( 4 \)[/tex] is less than or equal to [tex]\(4\)[/tex]. Thus, [tex]\( 4 \)[/tex] is included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = 4 \)[/tex] is in the domain of [tex]\( f(x) \)[/tex].
4. Consider [tex]\( x = 5 \)[/tex]:
- For the interval [tex]\( -6 < x \leq 0 \)[/tex]:
- [tex]\( 5 \)[/tex] is greater than 0. Thus, [tex]\( 5 \)[/tex] is not included in [tex]\( -6 < x \leq 0 \)[/tex].
- For the interval [tex]\( 0 < x \leq 4 \)[/tex]:
- [tex]\( 5 \)[/tex] is greater than [tex]\( 4 \)[/tex]. Thus, [tex]\( 5 \)[/tex] is not included in [tex]\( 0 < x \leq 4 \)[/tex].
- Therefore, [tex]\( x = 5 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
### Conclusion:
After checking each value, we find that the value [tex]\( x = 4 \)[/tex] is the only value that is in the domain of [tex]\( f(x) \)[/tex].
