Periodicity of Composite Functions

Composite functions are formed by combining two or more functions. If the functions involved in the composition are periodic, the resulting composite function may or may not be periodic. Let's consider the given options one by one to understand which of them is not always a nonperiodic function.

Option A

The composite function fog means first applying g and then f. If g is nonperiodic, then fog is also nonperiodic because the composition of a nonperiodic function and a periodic function is nonperiodic.

Option B

The composite function gof means first applying f and then g. If f is periodic and g is nonperiodic, then gof is not periodic, because the composition of a periodic function and a nonperiodic function is not periodic.

Option C

The composite function fof means applying f two times. If f is periodic, then fof is periodic because the composition of a periodic function with itself is periodic.

Option D

The composite function gog means applying g two times. If g is nonperiodic, then gog is also nonperiodic because the composition of a nonperiodic function with itself is nonperiodic.

Conclusion

From the above discussion, we can conclude that options A, B, and C are not always nonperiodic functions. Option A is not always nonperiodic because fog can be periodic if f is periodic. Option B is not always nonperiodic because gof can be periodic if f is periodic. Option C is not always nonperiodic because fof is periodic if f is periodic. Option D is always nonperiodic because gog is nonperiodic if g is nonperiodic.