To determine what value [tex]\( c \)[/tex] should be in order to keep the function one-to-one, we need to ensure that every [tex]\( x \)[/tex] value corresponds to a unique [tex]\( y \)[/tex] value. This means that no two [tex]\( x \)[/tex] values should map to the same [tex]\( y \)[/tex] value.
Given the points of the function:
[tex]\[
(1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, c)
\][/tex]
The [tex]\( y \)[/tex]-values from the known points are:
[tex]\[
2, 3, 5, 7, 11
\][/tex]
We are provided with several options for [tex]\( c \)[/tex]:
[tex]\[
2, 5, 11, 13
\][/tex]
To maintain the one-to-one nature of the function, we must choose a [tex]\( c \)[/tex] that is not already among the [tex]\( y \)[/tex]-values [tex]\(2, 3, 5, 7, 11\)[/tex].
Comparing the available options:
- [tex]\( 2 \)[/tex] is already in the [tex]\( y \)[/tex]-values.
- [tex]\( 5 \)[/tex] is already in the [tex]\( y \)[/tex]-values.
- [tex]\( 11 \)[/tex] is already in the [tex]\( y \)[/tex]-values.
- [tex]\( 13 \)[/tex] is not in the [tex]\( y \)[/tex]-values.
Therefore, the value of [tex]\( c \)[/tex] that will ensure the function remains one-to-one is:
[tex]\[
\boxed{13}
\][/tex]
