For what value of [tex]\( c \)[/tex] is the function one-to-one?

[tex]\(\{(1,2), (2,3), (3,5), (4,7), (5,11), (6,c)\}\)[/tex]

A. 2
B. 5
C. 11
D. 13



Answer :

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]