Question

If f(x) is a periodic function with principal period T then prove that the function f(ax + b) is periodic with principal period T/|a|.

Answer 1

If f(x) is a periodic function with principal period T then prove that the function f(ax + b)

[∴ T is a period of f(x)]

Thus, T/|a| | is a period of f(ax + b) ......(1)

Further, let a real number t > 0 be a period of f(ax + b). Then

f(ax + b) = f[a(x + t) + b], ∀x = f(ax + b + at), ∀x

⇒|a|t is a period of f(x)

⇒|a|t ≥ T (∴ T is the principal of f(x)]

⇒t ≥ T/|a|    .....(ii)

From Eqs. (1) and (2), it follows that T/|a|  is the principal period of f(ax + b).