Sure, let's analyze the set of ordered pairs given. The set of ordered pairs is [tex]\((1, 3), (2, 9), (k, 12), (5, 9), (8, 0)\)[/tex].
A function by definition maps each input (or x-value) to exactly one output (or y-value). Therefore, the x-values in the set of ordered pairs must be unique.
Let's identify the x-values from the given pairs:
- From (1, 3), the x-value is 1.
- From (2, 9), the x-value is 2.
- From (5, 9), the x-value is 5.
- From (8, 0), the x-value is 8.
These x-values (1, 2, 5, 8) are already present in the set. For the set to represent a function, the x-value 'k' in the pair [tex]\((k, 12)\)[/tex] must not duplicate any of the existing x-values.
Now let's check the multiple-choice options:
a) [tex]\( k = 0 \)[/tex]
- The value 0 is not among 1, 2, 5, or 8. Therefore, [tex]\( k \)[/tex] could be 0.
b) [tex]\( k = 1 \)[/tex]
- The value 1 is already an x-value in the pair (1, 3). Therefore, [tex]\( k \)[/tex] cannot be 1.
c) [tex]\( k = 5 \)[/tex]
- The value 5 is already an x-value in the pair (5, 9). Therefore, [tex]\( k \)[/tex] cannot be 5.
d) [tex]\( k = 8 \)[/tex]
- The value 8 is already an x-value in the pair (8, 0). Therefore, [tex]\( k \)[/tex] cannot be 8.
Given these analyses, the only value that [tex]\( k \)[/tex] could be without breaking the definition of a function is 0.
So, the correct answer is:
a) 0
