trigonometric functions

Question

What are the domain, range, and period of the function f(x) = –3cos(–2θ)?

Answer in complete sentences.

Answer 1

f(x) = –3cos(–2θ)

Domain is all θ.

Range is [-3, 3]

Period is 2π/2 = π

Answer 2

The domain is all real numbers, as we can replace θ with any real number and produce a valid output for the function.

The range of the function is [-3,3] (or in a different form -3 <= y <= 3). This comes from the fact that cos(θ) has a range [-1,1], and we are multiplying this by -3 (which flips the graph over the x-axis, and stretches it vertically by a factor of 3).

The -2 factor inside the cosine compresses the graph horizontally by a factor of 2, so we get a period of π (rather than the period of 2π that cosine usually has).