To determine the correct value of [tex]\( c \)[/tex] such that the function is one-to-one, we need to ensure that each [tex]\( x \)[/tex]-value maps to a unique [tex]\( y \)[/tex]-value.
One way to approach this problem is to observe the sequence of [tex]\( y \)[/tex]-values given by the function and identify any pattern. The given pairs are:
[tex]\[
(1, 2), (2, 3), (3, 5), (4, 7), (5, 11)
\][/tex]
First, let's determine if there is a discernible pattern in the [tex]\( y \)[/tex]-values:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 3 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y = 5 \)[/tex]
- For [tex]\( x = 4 \)[/tex], [tex]\( y = 7 \)[/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y = 11 \)[/tex]
Next, we compute the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
3 - 2 & = 1, \\
5 - 3 & = 2, \\
7 - 5 & = 2, \\
11 - 7 & = 4.
\end{align*}
\][/tex]
These differences do not follow a simple arithmetic progression. Instead, they suggest a pattern related to increments associated with prime numbers.
The provided sequence of [tex]\( y \)[/tex]-values appears to increase by the pattern of examining consecutive prime number differences. Specifically, let's test if the differences match the gaps between consecutive primes:
- [tex]\( 2 \to 3 \)[/tex] is the next prime after 2 (difference = 1)
- [tex]\( 3 \to 5 \)[/tex] (difference = 2)
- [tex]\( 5 \to 7 \)[/tex] (difference = 2)
- [tex]\( 7 \to 11 \)[/tex] (difference = 4)
The next prime after 11 is 13 which follows the pattern of prime numbers:
- The difference from 11 to 13 is [tex]\( 2 \)[/tex].
So, we retain the prime number sequence logic. The next number in this pattern after 11 is indeed 13.
Therefore, to maintain the one-to-one nature of the function, the value of [tex]\( c \)[/tex] should be:
[tex]\[
\boxed{13}
\][/tex]
