An oscillating voltage compressed horizontally is modeled by:Determine the domain, range, and period of the function if 0

Question

An oscillating voltage compressed horizontally is modeled by:Determine the domain, range, and period of the function if 0 < x < 2π

BlackTom AI · Accepted Answer

To begin, I restate the problem setup: we have y = sec(3x) and we consider the domain, range, and period for 0 < x < 2π. 
Option a is missing the actual domain and range details, showing placeholders; it provides no usable information, so it cannot be correct.
Option b proposes a domain: {x ∈ ℝ | x ≠ π/2, 3π/2, 5π/2, 7π/2, 9π/2} and a range {y ∈ ℝ | −1 ≤ y ≤ 1}, with period 2π. The domain listed here is inconsistent with y = sec(3x) because the zeros of cos(3x) occur at 3x = π/2 + kπ, i.e. x = π/6 + kπ/3, which yields a different set within 0 < x < 2π. Moreover, the range for sec is outside [−1, 1], not inside it, so this option misstates the range.
Option c gives the domain as {x ∈ ℝ | x ≠ π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6}, the range as {y ∈ ℝ | y ≤ −1 or y ≥ 1}, and the period as 2π/3. This aligns with the standard secant function behavior: the domain excludes points where cos(3x) = 0, namely x = π/6 + kπ/3, producing exactly the listed exclusions within 0 < x < 2π. The range of secant is y ≤ −1 or y ≥ 1, since sec is 1/cos and cosine ranges between −1 and 1, excluding zero at the asymptotes. The period of sec(kx) matches the period of cos(kx), which is 2π/k; here k = 3, so the period is 2π/3.
Option d lists a different, incorrect domain with many fractional multiples of π and claims a range {y ∈ ℝ | −1 ≤ y ≤ 1}, which again contradicts the known range of the secant function. It also lists a period of 2π/3, which is the right value, but since the domain and range components are incorrect, this option cannot be correct.
In summary, the only option that correctly describes all three components—domain excluding the correct set of points where cos(3x) = 0, range being y ≤ −1 or y ≥ 1, and period equal to 2π/3—is option c.

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

An oscillating voltage compressed horizontally is modeled by:Determine the domain, range, and period of the function if 0 < x < 2π

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