2. If a₁, a₂, a₃, a₄, a₅, and a₆ are six distinct real numbers and f(x) = |x²|, then the list f(a₁), f(a₂), f(a₃), f(a₄), f(a₅), and f(a₆) has a minimum of how many distinct values?
Answer: B
The list f(a₁), f(a₂), f(a₃), f(a₄), f(a₅), and f(a₆) has a minimum of 3 distinct values.
Given that f(x) = |x²|, we note that this function is non-negative and symmetric about the y-axis. Since the inputs are six distinct real numbers, the outputs will yield distinct values based on the squares of those numbers, leading to at least three distinct values when considering the possibility of both positive and negative inputs.
A) 2
This option is incorrect because the function f(x) = |x²| can produce more than two distinct values given six distinct inputs. The minimum number of distinct outputs must account for both positive and negative values of distinct inputs, which can create multiple unique results.
B) 3
This is the correct choice because with six distinct real numbers, it is reasonable to conclude that at least three distinct values will arise from the function f(x) = |x²|, as both positive and negative values can produce the same squared result, but variations in input ensure diversity in outputs.
C) 4
Option C is incorrect because while it is possible to have four distinct values, it is not the minimum. The presence of six distinct inputs guarantees at least three unique outputs, making this option less relevant to the question about the minimum.
D) 5
This option is incorrect as well. While it suggests a higher number of distinct values, it does not reflect the minimum condition required by the question. The minimum distinct outputs from the function f(x) = |x²| with six distinct inputs is three.
E) 6
This option is also incorrect because while having six distinct outputs is theoretically possible, it is not necessary. The question specifically asks for the minimum number of distinct values, which is three, thus making this option invalid.
Conclusion
The function f(x) = |x²|, when applied to six distinct real numbers, guarantees at least three distinct outputs due to the nature of squaring and the symmetry of the absolute value function. Options A, C, D, and E fail to align with the minimum requirement of distinct values, thus reinforcing that B is the only correct choice.