Which of the following is NOT a one-to-one function?

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BlackTom AI · Accepted Answer

To assess which option is NOT a one-to-one function, we examine whether each function passes the horizontal line test or has a strictly monotonic behavior.
Option a: f(x) = x^2. This parabola takes the same y-value for x and -x (e.g., f(2) = f(-2) = 4), so a horizontal line at y = 4 cuts the graph in two places. Therefore, it is not injective and not one-to-one.
Option b: f(x) = x^3 - 2. A cubic of this form is strictly increasing for all real x because its derivative f'(x) = 3x^2 is nonnegative and only zero at x = 0, but the function continues to increase overall. Consequently, each y corresponds to exactly one x, so it is one-to-one.
Option c: f(x) = 4x - 2. This is a linear function with a nonzero slope; it is strictly increasing across its entire domain, which guarantees a unique x for each y. Thus, it is one-to-one.
Option d: f(x) = 2/(x - 1) - 2. This rational function has domain x ≠ 1. Its derivative f'(x) = -2/(x - 1)^2 is negative for all x ≠ 1, meaning the function is strictly decreasing on its entire domain. A strictly decreasing function is injective, so it is one-to-one.
In summary, the parabola in option a fails the horizontal line test and is not one-to-one, while options b, c, and d preserve one-to-one behavior through monotonicity or injectivity on their domains.

Which of the following is NOT a one-to-one function?

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