To determine which number could replace [tex]\( N \)[/tex] so that the table represents a function, we need to ensure that the table adheres to the definition of a function. Specifically, each input (or [tex]\( x \)[/tex]-value) must map to exactly one output (or [tex]\( y \)[/tex]-value). This means that no two rows in the table should have the same [tex]\( x \)[/tex]-value.
We can start by identifying the [tex]\( x \)[/tex]-values already present in the table.
The [tex]\( x \)[/tex]-values given are:
- [tex]\( x = 12 \)[/tex]
- [tex]\( x = 9 \)[/tex]
- [tex]\( x = 4 \)[/tex]
- [tex]\( x = 11 \)[/tex]
Now, we have to choose a value for [tex]\( N \)[/tex] such that it does not repeat any of these [tex]\( x \)[/tex]-values. The possible values provided for [tex]\( N \)[/tex] are:
- [tex]\( N = 11 \)[/tex]
- [tex]\( N = 12 \)[/tex]
- [tex]\( N = 4 \)[/tex]
Let's analyze each possibility:
1. If [tex]\( N = 11 \)[/tex]:
The [tex]\( x \)[/tex]-values would be [tex]\( \{12, 9, 4, 11, 11\} \)[/tex]. This set has a repeat of the number 11, thus violating the definition of a function.
2. If [tex]\( N = 12 \)[/tex]:
The [tex]\( x \)[/tex]-values would be [tex]\( \{12, 9, 4, 11, 12\} \)[/tex]. This set has a repeat of the number 12, thus violating the definition of a function.
3. If [tex]\( N = 4 \)[/tex]:
The [tex]\( x \)[/tex]-values would be [tex]\( \{12, 9, 4, 11, 4\} \)[/tex]. This set has a repeat of the number 4, thus violating the definition of a function.
Since all provided options for [tex]\( N \)[/tex] repeat an existing [tex]\( x \)[/tex]-value in the table, there is no possible number from the given choices (11, 12, or 4) that can be used to replace [tex]\( N \)[/tex] without violating the one-to-one mapping required for a set of ordered pairs to represent a function.
Hence, no number from the given options can replace [tex]\( N \)[/tex] to make the table represent a function.
