Which piecewise relation defines a function?

A. [tex] y = \left\{ \begin{aligned} x^2, & \quad x \ \textless \ -2 \\ 0, & \quad -2 \leq x \leq 4 \\ -x^2, & \quad x \geq 4 \end{aligned} \right. [/tex]

B. [tex] y = \left\{ \begin{array}{ll} x^2, & x \leq -2 \\ 4, & -2 \ \textless \ x \leq 2 \\ x^2 + 1, & x \geq 2 \end{array} \right. [/tex]

C. [tex] y = \left\{ \begin{array}{ll} -3x, & x \ \textless \ -2 \\ 3, & 0 \leq x \ \textless \ 4 \\ 2x, & x \geq 4 \end{array} \right. [/tex]

D. [tex] y = \left\{ \begin{aligned} -3x, & x \leq -4 \\ 3, & -5 \ \textless \ x \ \textless \ 1 \end{aligned} \right. [/tex]



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].