The derivatives of trigonometric functions consist of sinx, cosx,tanx, cotx, secx, and cscx. All these functions are continuous and differentiable in their _______.

Question

Choices:

A. domains

B. ranges

C. intervals

D. x and y – axes

Answer 1

The answer is A, domains.

To check a function for continuity and differentiability, we want to know how it behaves when we plug in different values of x. The function tan(x), for instance, is not defined when x=(pi)/2+(pi)k where k is an integer. The term describing the values of x that a function can take is called the domain, so it only makes sense to speak of a function being defined/continuous/differentiable within its domain.

To see why it cannot be B, for instance, take sin(x) and cos(x), which both have a range (interval on which the functions' outputs range) of [-1, 1]. These functions are continuous and differentiable well out of this interval, since they will give you a (differentiable) output for any value of x, not just the ones in the interval [-1, 1].