Question

A real-valued function f is said to be periodic with period T ≠ 0 if f(x + T) = f(x) for all x in the domain of f. If T is the smallest positive value for which f(x + T) = f(x) holds, then T is called the fundamental period of f. Determine the fundamental period T of the given function. f(x) = cos(2πx)

Answer 1

f(x) = cos 2πx

Thus the fundamental period T = 1

Explanation:

From the information given:

The periodic function f(x + T) = f(x) T ≠ 0 provided that T is the smallest positive value for which f(x + T) = f(x) holds, then T is called the fundamental period of f.

Now, for the given function f(x) = cos(2πx)

suppose T = 1/4 since T ≠ 0

Then:

f(x+ 1/4) = cos(2π (x + 1/4) )

f(x+ 1/4) = cos (2πx + π/2)

f(x+ 1/4) = -sin 2πx (Not satisfied)

suppose T = 1/2 since T ≠ 0

Then:

f(x+ 1/2) = cos(2π (x + 1/2) )

f(x+ 1/2) = cos (2πx + π)

f(x+ 1/2) = -cos 2πx (Not satisfied)

suppose T = 1 since T ≠ 0

Then:

f(x+ 1) = cos(2π (x + 1) )

f(x+ 1) = cos (2πx + 2π)

f(x+ 1) = cos 2πx

f(x+ 1) = f(x)

f(x) = cos 2πx

Therefore, the fundamental period T = 1 since f(x) = cos 2πx