Questions: Choose the graph of y=sec x. Answer (a) and (b). (a) Is the function an even function? yes no (b) Does the function have a period of P=2 π ? no yes

Solution Steps

The secant function, sec(x), is the reciprocal of the cosine function, cos(x). It has vertical asymptotes where cos(x) = 0, which are at odd multiples of π/2. The second graph is the correct graph of y = sec x.

An even function satisfies f(-x) = f(x) for all x in the domain. Since sec(x) = 1/cos(x) and cos(-x) = cos(x), it follows that sec(-x) = 1/cos(-x) = 1/cos(x) = sec(x). Thus, the secant function is an even function.

The period of a function is the smallest positive value P such that f(x + P) = f(x) for all x. The cosine function has a period of 2π, meaning cos(x + 2π) = cos(x) for all x. Since sec(x) = 1/cos(x), it follows that sec(x + 2π) = 1/cos(x + 2π) = 1/cos(x) = sec(x). So, the secant function has a period of 2π.

Final Answer: