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

Question

BlackTom AI · Accepted Answer

Let's restate the problem in our own words to ensure clarity: we are asked to identify which option does not describe a one-to-one (injective) function.

Option A: f: R^+ → R, f(x) = log_e(x).
- The natural logarithm is strictly increasing on the positive real numbers, and each distinct input x>0 gives a distinct output log(x). Therefore, this function is one-to-one.
- A common misconception would be to worry about the codomain being all of R, but injectivity only concerns whether different inputs produce different outputs; the codomain does not affect injectivity here.

Option B: f: R → R, f(x) = sin(x).
- The sine function is periodic with period 2π, so many different x values yield the same sine value. For example, sin(0) = sin(2π) = 0. This repetition shows the function is not injective.
- A frequent pitfall is to assume that trigonometric functions must be one-to-one on their entire real domain; however, sin(x) is not one-to-one over all real numbers due to its repeating pattern.

Option C: f: R → R, f(x) = 2x (and the notation also shows f(x) = 2^x elsewhere).
- If the function is f(x) = 2x, this is a linear function with slope 2, which maps distinct x to distinct values, so it is one-to-one.
- If instead the intended function were f(x) = 2^x, the exponential function is also strictly increasing on R and thus injective, even though its codomain is R+; it remains one-to-one when viewed as a function into R (since its outputs are still distinct for distinct inputs).
- A potential confusion arises from the mixed notation in the option, but in either standard interpretation (2x or 2^x), the function remains injective.

Option D: f: R → R, f(x) = x^3.
- The cube function is strictly increasing on all of R, so different inputs produce different outputs. It is injective.
- A misunderstanding might come from thinking about odd vs even powers, but injectivity depends on monotonicity over the entire domain, which x^3 possesses.

In summary, examining each option reveals that A, C, and D define injective (one-to-one) functions under their typical interpretations, while B fails the injectivity criterion due to periodicity and repetition of output values for different inputs.

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

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