Answer :
To determine which piecewise relations define a function, we need to verify that each relation assigns exactly one value of [tex]\( y \)[/tex] for each value of [tex]\( x \)[/tex]. Let's examine each piecewise relation one by one.
### 1. First Relation:
[tex]\[ y = \left\{ \begin{aligned} &x^2, \quad &x < -2 \\ &0, \quad &-2 \leq x \leq 4 \\ &-x^2, \quad &x \geq 4 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x < -2 \)[/tex], [tex]\( y = x^2 \)[/tex]. This is a well-defined function.
- For [tex]\( -2 \leq x \leq 4 \)[/tex], [tex]\( y = 0 \)[/tex]. This is a constant function and well-defined.
- For [tex]\( x \geq 4 \)[/tex], [tex]\( y = -x^2 \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 2. Second Relation:
[tex]\[ y = \left\{ \begin{aligned} &x^2, \quad &x \leq -2 \\ &4, \quad &-2 < x \leq 2 \\ &x^2 + 1, \quad &x \geq 2 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x \leq -2 \)[/tex], [tex]\( y = x^2 \)[/tex]. This is a well-defined function.
- For [tex]\( -2 < x \leq 2 \)[/tex], [tex]\( y = 4 \)[/tex]. This is constant and well-defined.
- For [tex]\( x \geq 2 \)[/tex], [tex]\( y = x^2 + 1 \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 3. Third Relation:
[tex]\[ y = \left\{ \begin{aligned} &-3x, \quad &x < -2 \\ &3, \quad &0 \leq x < 4 \\ &2x, \quad &x \geq 4 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x < -2 \)[/tex], [tex]\( y = -3x \)[/tex]. This is a well-defined function.
- For [tex]\( 0 \leq x < 4 \)[/tex], [tex]\( y = 3 \)[/tex]. This is constant and well-defined.
- For [tex]\( x \geq 4 \)[/tex], [tex]\( y = 2x \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 4. Fourth Relation:
[tex]\[ y = \left\{ \begin{aligned} &-3x, \quad &x \leq -4 \\ &3, \quad &-5 < x < 1 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x \leq -4 \)[/tex], [tex]\( y = -3x \)[/tex]. This is a well-defined function.
- For [tex]\( -5 < x < 1 \)[/tex], [tex]\( y = 3 \)[/tex]. This is constant and well-defined.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### Conclusion
All four piecewise relations define a function. Hence, the piecewise relations that define a function are labeled as numbers [tex]\(1, 2, 3,\)[/tex] and [tex]\(4\)[/tex].
### 1. First Relation:
[tex]\[ y = \left\{ \begin{aligned} &x^2, \quad &x < -2 \\ &0, \quad &-2 \leq x \leq 4 \\ &-x^2, \quad &x \geq 4 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x < -2 \)[/tex], [tex]\( y = x^2 \)[/tex]. This is a well-defined function.
- For [tex]\( -2 \leq x \leq 4 \)[/tex], [tex]\( y = 0 \)[/tex]. This is a constant function and well-defined.
- For [tex]\( x \geq 4 \)[/tex], [tex]\( y = -x^2 \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 2. Second Relation:
[tex]\[ y = \left\{ \begin{aligned} &x^2, \quad &x \leq -2 \\ &4, \quad &-2 < x \leq 2 \\ &x^2 + 1, \quad &x \geq 2 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x \leq -2 \)[/tex], [tex]\( y = x^2 \)[/tex]. This is a well-defined function.
- For [tex]\( -2 < x \leq 2 \)[/tex], [tex]\( y = 4 \)[/tex]. This is constant and well-defined.
- For [tex]\( x \geq 2 \)[/tex], [tex]\( y = x^2 + 1 \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 3. Third Relation:
[tex]\[ y = \left\{ \begin{aligned} &-3x, \quad &x < -2 \\ &3, \quad &0 \leq x < 4 \\ &2x, \quad &x \geq 4 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x < -2 \)[/tex], [tex]\( y = -3x \)[/tex]. This is a well-defined function.
- For [tex]\( 0 \leq x < 4 \)[/tex], [tex]\( y = 3 \)[/tex]. This is constant and well-defined.
- For [tex]\( x \geq 4 \)[/tex], [tex]\( y = 2x \)[/tex]. This is a well-defined function.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### 4. Fourth Relation:
[tex]\[ y = \left\{ \begin{aligned} &-3x, \quad &x \leq -4 \\ &3, \quad &-5 < x < 1 \end{aligned} \right. \][/tex]
For this relation:
- For [tex]\( x \leq -4 \)[/tex], [tex]\( y = -3x \)[/tex]. This is a well-defined function.
- For [tex]\( -5 < x < 1 \)[/tex], [tex]\( y = 3 \)[/tex]. This is constant and well-defined.
Thus, this piecewise relation defines a function as each interval provides exactly one [tex]\( y \)[/tex] value for any [tex]\( x \)[/tex] in that interval.
### Conclusion
All four piecewise relations define a function. Hence, the piecewise relations that define a function are labeled as numbers [tex]\(1, 2, 3,\)[/tex] and [tex]\(4\)[/tex].
