Answer :
To determine which [tex]\( y \)[/tex]-values are within the range of the given piecewise function:
[tex]\[ f(x)=\left\{\begin{array}{ll} 2 x + 2, & \text{if } x < -3 \\ x, & \text{if } x = -3 \\ -x - 2, & \text{if } x > -3 \end{array}\right. \][/tex]
We will check each given [tex]\( y \)[/tex]-value to determine if it falls within any segment of the function.
### Checking [tex]\( y = -6 \)[/tex]
For [tex]\( y = -6 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -6 \implies 2x = -8 \implies x = -4 \][/tex]
Since [tex]\( x = -4 \)[/tex] and [tex]\(-4 < -3\)[/tex], [tex]\( y = -6 \)[/tex] is in the range in this segment.
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -6 \quad (\text{but this is not within the domain of \( x = -3 \)}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -6 \implies -x = -4 \implies x = 4 \][/tex]
Since [tex]\( x = 4 \)[/tex] and [tex]\(4 > -3\)[/tex], [tex]\( y = -6 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = -6 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = -4 \)[/tex]
For [tex]\( y = -4 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -4 \implies 2x = -6 \implies x = -5 \][/tex]
Since [tex]\( x = -5 \)[/tex] and [tex]\(-5 < -3\)[/tex], [tex]\( y = -4 \)[/tex] is in the range in this segment.
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -4 \quad (\text{but this is not within the domain of \( x = -3 \)}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -4 \implies -x = -2 \implies x = 2 \][/tex]
Since [tex]\( x = 2 \)[/tex] and [tex]\(2 > -3\)[/tex], [tex]\( y = -4 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = -4 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = -3 \)[/tex]
For [tex]\( y = -3 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -3 \implies 2x = -5 \implies x = -\frac{5}{2} \quad (\text{but } -\frac{5}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -3 \quad (\text{this exactly matches the domain of } x = -3) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -3 \implies -x = -1 \implies x = 1 \quad (\text{but } 1 > -3, none of the conditions are aligned) \][/tex]
Hence, [tex]\( y = -3 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 0 \)[/tex]
For [tex]\( y = 0 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 0 \implies 2x = -2 \implies x = -1 \quad (\text{but } -1 \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad (\text{but this is not within the domain of } x = -3 \text{, excluded}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 0 \implies -x = 2 \implies x = -2 \][/tex]
Since [tex]\( x = -2 \)[/tex] and [tex]\(-2 > -3\)[/tex], [tex]\( y = 0 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = 0 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 1 \)[/tex]
For [tex]\( y = 1 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 1 \implies 2x = -1 \implies x = -\frac{1}{2} \quad (\text{but } -\frac{1}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = 1 \quad (\text{ not matching within domain } x=-3) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 1 \implies -x = 3 \implies x = -3 \quad (\text{but this is not satisfying the positivity}) \][/tex]
Hence, [tex]\( y = 1 \)[/tex] is not within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 3 \)[/tex]
For [tex]\( y = 3 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 3 \implies 2x = 1 \implies x = \frac{1}{2} \quad (\text{but } \frac{1}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x= 3 \quad (\text{not satisfying the equality} \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 3 \implies -x = 5 \implies x=-5 \quad (\text{but not found in permissible domain}) \][/tex]
Hence, [tex]\( y = 3 \)[/tex] is not within the range of [tex]\( f(x) \)[/tex].
### Summary
Based on the detailed check above, the values that lie within the range of the function [tex]\( f(x) \)[/tex] are:
[tex]\[ \boxed{-6, -4, -3, 0} \][/tex]
[tex]\[ f(x)=\left\{\begin{array}{ll} 2 x + 2, & \text{if } x < -3 \\ x, & \text{if } x = -3 \\ -x - 2, & \text{if } x > -3 \end{array}\right. \][/tex]
We will check each given [tex]\( y \)[/tex]-value to determine if it falls within any segment of the function.
### Checking [tex]\( y = -6 \)[/tex]
For [tex]\( y = -6 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -6 \implies 2x = -8 \implies x = -4 \][/tex]
Since [tex]\( x = -4 \)[/tex] and [tex]\(-4 < -3\)[/tex], [tex]\( y = -6 \)[/tex] is in the range in this segment.
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -6 \quad (\text{but this is not within the domain of \( x = -3 \)}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -6 \implies -x = -4 \implies x = 4 \][/tex]
Since [tex]\( x = 4 \)[/tex] and [tex]\(4 > -3\)[/tex], [tex]\( y = -6 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = -6 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = -4 \)[/tex]
For [tex]\( y = -4 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -4 \implies 2x = -6 \implies x = -5 \][/tex]
Since [tex]\( x = -5 \)[/tex] and [tex]\(-5 < -3\)[/tex], [tex]\( y = -4 \)[/tex] is in the range in this segment.
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -4 \quad (\text{but this is not within the domain of \( x = -3 \)}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -4 \implies -x = -2 \implies x = 2 \][/tex]
Since [tex]\( x = 2 \)[/tex] and [tex]\(2 > -3\)[/tex], [tex]\( y = -4 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = -4 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = -3 \)[/tex]
For [tex]\( y = -3 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = -3 \implies 2x = -5 \implies x = -\frac{5}{2} \quad (\text{but } -\frac{5}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = -3 \quad (\text{this exactly matches the domain of } x = -3) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = -3 \implies -x = -1 \implies x = 1 \quad (\text{but } 1 > -3, none of the conditions are aligned) \][/tex]
Hence, [tex]\( y = -3 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 0 \)[/tex]
For [tex]\( y = 0 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 0 \implies 2x = -2 \implies x = -1 \quad (\text{but } -1 \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad (\text{but this is not within the domain of } x = -3 \text{, excluded}) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 0 \implies -x = 2 \implies x = -2 \][/tex]
Since [tex]\( x = -2 \)[/tex] and [tex]\(-2 > -3\)[/tex], [tex]\( y = 0 \)[/tex] is in the range in this segment.
Hence, [tex]\( y = 0 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 1 \)[/tex]
For [tex]\( y = 1 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 1 \implies 2x = -1 \implies x = -\frac{1}{2} \quad (\text{but } -\frac{1}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x = 1 \quad (\text{ not matching within domain } x=-3) \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 1 \implies -x = 3 \implies x = -3 \quad (\text{but this is not satisfying the positivity}) \][/tex]
Hence, [tex]\( y = 1 \)[/tex] is not within the range of [tex]\( f(x) \)[/tex].
### Checking [tex]\( y = 3 \)[/tex]
For [tex]\( y = 3 \)[/tex]:
1. Segment [tex]\( 2x + 2 \)[/tex]:
[tex]\[ 2x + 2 = 3 \implies 2x = 1 \implies x = \frac{1}{2} \quad (\text{but } \frac{1}{2} \text{ is not less than } -3) \][/tex]
2. Segment [tex]\( x \)[/tex]:
[tex]\[ x= 3 \quad (\text{not satisfying the equality} \][/tex]
3. Segment [tex]\( -x - 2 \)[/tex]:
[tex]\[ -x - 2 = 3 \implies -x = 5 \implies x=-5 \quad (\text{but not found in permissible domain}) \][/tex]
Hence, [tex]\( y = 3 \)[/tex] is not within the range of [tex]\( f(x) \)[/tex].
### Summary
Based on the detailed check above, the values that lie within the range of the function [tex]\( f(x) \)[/tex] are:
[tex]\[ \boxed{-6, -4, -3, 0} \][/tex]
