Which statement best describes the period of ?

Question

BlackTom AI · Accepted Answer

Let's break down what is meant by the period of a function like f(x) = tan(x).
Option a: π/2. This would imply the function repeats every π/2 units. However, tan(x) has vertical asymptotes at x = π/2 + kπ, and its basic repeating pattern occurs over an interval of length π, not π/2. So π/2 cannot be the fundamental period.
Option b: π. This is the standard period of the tangent function. The tangent function satisfies tan(x + π) = tan(x) for all x where both sides are defined, meaning its pattern repeats every π along the x-axis. This matches the graph’s repeating cycle between consecutive asymptotes, which are π apart.
Option c: ∞. Saying the period is infinite would mean the function never repeats, which is false for tan(x). It has a clear, regular repetition every π.
Option d: 2π. While 2π is a multiple of the fundamental period, the smallest positive amount that repeats tan(x) is π, not 2π. So 2π is not the fundamental period, though the function does repeat again after 2π as well.
In summary, the period concept asks for the smallest positive length T such that tan(x + T) = tan(x) wherever both sides are defined. The only option that matches this property as the smallest such length is π.

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

Which statement best describes the period of ?

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