To determine whether [tex]\( f(x) \)[/tex] is a function, we must verify if each input value [tex]\( x \)[/tex] maps to exactly one output value [tex]\( f(x) \)[/tex]. In other words, for every unique [tex]\( x \)[/tex], there must be a unique and consistent [tex]\( f(x) \)[/tex].
Let's go through the given values step-by-step:
1. We are given the input values [tex]\( x \)[/tex] as follows: [tex]\( 0, 4, 82, 12, 3, 2, 0 \)[/tex].
2. We are also given the output values [tex]\( f(x) \)[/tex] as follows: [tex]\( 4, 82, 12, 3, 2, 0 \)[/tex].
Next, we check if each input [tex]\( x \)[/tex] corresponds to a unique [tex]\( f(x) \)[/tex]:
- The input [tex]\( x = 0 \)[/tex] appears twice in the list of inputs.
When [tex]\( x = 0 \)[/tex] appears more than once, for [tex]\( f(x) \)[/tex] to be qualified as a function, the corresponding output [tex]\( f(x) \)[/tex] for each occurrence of [tex]\( x = 0 \)[/tex] must be the same. However, in this problem setup, we are more concerned about whether any input [tex]\( x \)[/tex] is repeated in the set of inputs.
Since we have repeated values of [tex]\( x = 0 \)[/tex]:
- This repetition indicates that not every input [tex]\( x \)[/tex] has a unique mapping, or it raises a suspicion regarding the output mappings corresponding to the repeated input.
Consequently, with the presence of a duplicate input value [tex]\( x = 0 \)[/tex] in the list, we cannot guarantee that [tex]\( f(x) \)[/tex] is a function.
Thus, based on the given values, the answer to whether [tex]\( f(x) \)[/tex] is a function is:
A. False.
Therefore, the true result is [tex]\( \boxed{A} \)[/tex].
