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

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

Option a: f(x) = 2/(x-1) - 2. This is a rational function with a vertical asymptote at x = 1. Although it is not defined at x = 1, we can check injectivity by solving f(x1) = f(x2). From 2/(x1-1) - 2 = 2/(x2-1) - 2, we get 2/(x1-1) = 2/(x2-1), which implies x1-1 = x2-1, hence x1 = x2. Therefore, no two distinct x give the same y, so this function is injective on its domain.

Option b: f(x) = 4x - 2. This is a linear function with constant slope 4. A linear function with nonzero slope is injective because if f(x1) = f(x2), then 4x1 - 2 = 4x2 - 2, leading to x1 = x2. Thus it is one-to-one.

Option c: f(x) = x^2. Here, f(2) = 4 and f(-2) = 4, so two different inputs yield the same output. This is the classic example of a non-injective (not one-to-one) behavior when considering all real numbers as the domain. The presence of paired inputs x and -x mapping to the same value demonstrates the violation of injectivity.

Option d: f(x) = x^3 - 2. The cubic x^3 is strictly increasing over all real numbers, and subtracting 2 shifts the graph down but does not affect monotonicity. A strictly increasing function is injective, since f(x1) = f(x2) would force x1 = x2. Therefore, this option is one-to-one.

Summary: Each option was evaluated for injectivity on its domain. The only choice where two distinct inputs can produce the same output is option c, f(x) = x^2, which makes it NOT a one-to-one function. The other functions maintain distinct outputs for distinct inputs across their domains.

MUF0091 Mathematics Unit 1 - Semester 1, 2025 3.6 Power functions review questions

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

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