Final answer:
Trigonometric functions can have different domain restrictions to remain one-to-one. It's essential that these restrictions align with a single, non-repeating cycle of the function. Changing the periodic domain improperly can invalidate the one-to-one nature of these functions.
Step-by-step explanation:
Trigonometric functions are periodic and, in their original form, are not one-to-one. To make trigonometric functions one-to-one, we can apply domain restrictions. For example, the sine function is one-to-one when restricted to the interval [-π/2, π/2], and the cosine function is one-to-one when restricted to [0, π]. The key is to select a domain where the function doesn't repeat its values, also known as its principal domain. Different domain restrictions can be applied to a trig function to keep it one-to-one, as long as the restricted domain covers exactly one cycle of the trigonometric function without overlap.
As a practical example, consider the tangent function which is naturally one-to-one within each interval of π (pi) such as (-π/2, π/2). Another domain restriction like (-π, π) will also keep the tangent function one-to-one. Changing the period or applying non-standard domain restrictions may invalidate the one-to-one nature if the new domain includes repeating values of the function.