For the tangent function, what values of x must be excluded from its domain?

The tangent function has specific points where it is undefined due to its relationship with the sine and cosine functions. The tangent function can be expressed as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). The function is undefined whenever the denominator (cos(x)) is equal to zero.

The cosine function equals zero at all odd multiples of π/2, which can be represented as (2n + 1)(π)/2, where n is any integer (n = 0, ±1, ±2, ...). At these values, the tangent function will have vertical asymptotes, meaning the output is undefined.

Therefore, the correct response regarding which values of x must be excluded from the domain of the tangent function is indeed related to these points where cosine is zero, confirming that x cannot equal (2n + 1)(π)/2. This is why this choice accurately identifies the restrictions on the domain for the tangent function.