Final answer:
The period of the function f(x) = sin^2 x + cos x is 2π, as it is the least common multiple of the periods of sin x and cos x. The function has no asymptotes because both sin^2 x and cos x are bounded functions without discontinuities. The range of the function is from -1 to 2.
Step-by-step explanation:
The question pertains to finding the characteristics of a sinusoidal wave based on a given function f(x) = sin^2 x + cos x.
The period of the function can be determined by looking at the individual periods of sin x and cos x, which are both 2π. Since sin^2 x is just sin x squared, its period is also 2π. Therefore, the period of the combined function f(x) is 2π, which is the least common multiple of the periods of its constituent functions.
There are no asymptotes for this function because sin^2 x and cos x are both bounded functions, and their sum does not introduce any discontinuities or divisions by zero.
The range of the function is the set of all possible output values. sin^2 x varies between 0 and 1, and cos x varies between -1 and 1. At the minimum, when both functions are at their lowest, f(x) is -1. At the maximum, when sin^2 x is at 1 and cos x is at 1, f(x) reaches 2. Therefore, the range of f(x) = sin^2 x + cos x is from -1 to 2.