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?

Rationale

At least 3 distinct values can be obtained from the function f(x) = |x²|.

The function f(x) = |x²| is non-negative, and since a₁, a₂, a₃, a₄, a₅, and a₆ are distinct real numbers, the outputs of f will depend on the distinct squares of these numbers. The minimum number of distinct values occurs when two or more of the inputs yield the same square.

A (2) YOUR ANSWER

If there were only 2 distinct values, it would imply that at least four of the distinct numbers would have to produce the same output from the function, which is not possible given that the inputs are distinct. Thus, it is not feasible for the function to yield just two distinct values from six different inputs.

B (3) CORRECT

This is the correct answer because it is possible for three distinct values to arise from the function when we consider both positive and negative numbers. For example, if we choose three positive numbers and their corresponding negative equivalents, the squares will overlap, leading to three distinct values: zero (from f(0)), and two positive squares from the remaining numbers.

C (4) YOUR ANSWER

While it is possible to have four distinct values, it is not a minimum. To achieve four distinct values, we would need to ensure that all four numbers produce different squares without overlaps, which is not the least number achievable given the distinct nature of the inputs.

D (5) YOUR ANSWER

Similar to option C, achieving five distinct values would require careful selection of the six distinct numbers to ensure that exactly one pair produces the same square. This scenario is not a minimum and exceeds what can be obtained with distinct inputs.

E (6) YOUR ANSWER

This option suggests that all six distinct numbers produce unique outputs, which is not feasible since pairs of negative and positive numbers would yield the same square, thus reducing the total number of distinct outputs below six.

Conclusion

The function f(x) = |x²| applied to six distinct real numbers will yield at least three distinct values, as overlapping squares from negative and positive pairs allow for such a minimum. While more distinct values can be achieved depending on the specific selections of the inputs, the scenario of three distinct values is the minimum that can be reasonably expected.

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?

At least 3 distinct values can be obtained from the function f(x) = |x²|.

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