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

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BlackTom AI · Accepted Answer

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

Option a: f(x) = x^3 - 2. A cubic with odd degree and monotonic increasing derivative overall; since x^3 is strictly increasing, subtracting 2 shifts the graph down but does not create any duplicates in y for different x. Therefore this is one-to-one.

Option b: f(x) = x^2. This is the classic parabola opening upward. It is not one-to-one because f(1) = f(-1) = 1, so two different inputs yield the same output.

Option c: f(x) = 4x - 2. This is a linear function with nonzero slope. Linear functions with nonzero slope are always one-to-one because they pass the horizontal line test: each y corresponds to exactly one x.

Option d: f(x) = 2/(x-1) - 2. This is a rational function with a vertical asymptote at x = 1. On its domain (all real numbers except 1), the function is strictly monotone (decreasing on each interval separated by the asymptote) and, importantly, the two branches do not produce the same y-value for different x across the domain. Hence it is one-to-one on its domain.

Summary of reasoning:
- A function is not one-to-one if there exist at least two distinct inputs giving the same output. This happens for the parabola x^2 because of symmetry about the y-axis.
- The cubic x^3 - 2 and the linear 4x - 2 are strictly monotone over their entire domains, so they are one-to-one.
- The rational function 2/(x-1) - 2 has a vertical asymptote and two branches that do not share outputs for different inputs within the domain, making it one-to-one on its domain.

Therefore, the option that fails the one-to-one test is the parabola f(x) = x^2.

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

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Read the Horizontal Line Test in the textbook Links to an external site. . Which of the following statements are equivalent to (i.e. say the same thing as) the Horizontal Line Test? Select all that apply.