Explanation:
To understand why options A, B, and C are not always nonperiodic functions, we need to consider the properties of periodic and nonperiodic functions.

Periodic Function:
A function f(x) is said to be periodic if there exists a positive number T such that f(x+T) = f(x) for all values of x. T is called the period of the function.

Nonperiodic Function:
A function g(x) is said to be nonperiodic if there is no positive number T such that g(x+T) = g(x) for all values of x.

Now, let's analyze each option individually:

a) fog:
When we take the composition of a periodic function f(x) and a nonperiodic function g(x), the resulting function fog(x) may or may not be periodic. It depends on the specific functions f and g. If the composition fog(x) is periodic, then it is not a nonperiodic function. Therefore, option A is not always a nonperiodic function.

b) gof:
Similarly, when we take the composition of a nonperiodic function g(x) and a periodic function f(x), the resulting function gof(x) may or may not be periodic. If the composition gof(x) is periodic, then it is not a nonperiodic function. Hence, option B is not always a nonperiodic function.

c) fof:
In this case, we are taking the composition of a periodic function f(x) with itself. The resulting function fof(x) will always be periodic because the period of fof(x) will be the same as the period of f(x). Therefore, option C is not always a nonperiodic function.

d) gog:
Here, we are taking the composition of a nonperiodic function g(x) with itself. Since g(x) is nonperiodic, the composition gog(x) will also be nonperiodic. Therefore, option D is always a nonperiodic function.

To summarize, options A, B, and C are not always nonperiodic functions, but option D is always a nonperiodic function.