What is the range of the secant function?

The secant function, defined as the reciprocal of the cosine function, demonstrates a specific range based on the values that the cosine function can take. Since the cosine function has a range of [-1, 1], the secant function, being the reciprocal of cosine, has notable characteristics.

When the cosine function is at its maximum of 1, the secant function equals 1, leading to the lowest value in the range. However, when the cosine function approaches 0 (which occurs at odd multiples of π/2), the secant function approaches positive or negative infinity, indicating that there are no secant values between -1 and 1, where cosine outputs zeros or values that are too close to zero. Thus, the secant function can take on all values greater than or equal to 1 (from the positive side) and all values less than or equal to -1 (from the negative side).

This analysis directly leads to the conclusion that the range of the secant function is [1, ∞) in union with (-∞, -1], confirming that the correct answer is indeed the range option stated.