For which interval is f(x) = \dfrac{1}{x - 2} negative?

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The function given is f(x) = 1/(x - 2). First, note its domain excludes x = 2 because the denominator would be zero there.

- Option a: (2, ∞). In this interval, x - 2 > 0, so f(x) = 1/(positive) is positive, not negative. This makes this interval incorrect.

- Option b: (-∞, 1). This interval is entirely to the left of 1, which is still left of 2, so in this region x - 2 < 0 and f(x) would be negative. However, this option does not include all x < 2; it stops at 1 and omits the portion (1, 2). Therefore, it does not represent all x where f(x) is negative, so it is not correct.

- Option c: (-∞, 2). Here every x less than 2 yields x - 2 < 0, so f(x) = 1/(x - 2) is negative throughout this entire interval. This matches the set of all x for which f(x) is negative, making this option correct.

- Option d: (-∞, ∞). This includes x = 2 (where the function is undefined) and also includes values greater than 2 where the function is positive. Since the interval includes points where the function is not negative, this choice is incorrect.

In summary, because the sign of f(x) is determined by the sign of x - 2 (with the numerator fixed at 1), the function is negative exactly for all x < 2, i.e., on the interval (-∞, 2). The other options either miss portions of this interval or include regions where f(x) is not negative.

MHF4U - Advanced Functions 12 (2025-26) - A

For which interval is f(x) = \dfrac{1}{x - 2} negative?

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