Investigating Functions
Let be a periodic function and be a non-periodic function, which of the following statement(s) is/are true?
Statement 1
:
will be periodic.
Statement 2
:
will be periodic.
Statement 3
: Both
and
maybe periodic.
Statement 4
: All the statements above are true.
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1 solution
if f(x) is periodic and g(x) is non periodic then g(f(x)) will be periodic but f(g(x)) may or may not be periodic. Generally in case of composite function, function will be periodic when inner function is periodic. for example, if g(x)= x^(2); and f(x)= cos(x);then gof(x)=[cos(x)]^2 is periodic with period (pi), but fog(x)=cos(x^2) will be non periodic.
but if f(x) is periodic and g(x) is a linear function of x.then both fog(x) and gof(x) will be periodic. for example f(x)=sin(x); and g(x)= ax+b; then fog(x)=sin(ax+b) is periodic with period [2 pi/mod(a)], also gof(x)=a(sin x)+b is periodic with period (2 pi).