Q) If f(x) is symmetrical about the line x = a, x = b (a b) - Maths - Relations and Functions

Question

Q) If f(x) is symmetrical about the line x = a, x = b (a > b)
Prove that f(x) is periodic

Aarushi Mishra · Accepted Answer

If fx is symmetric about x=a thenfa+x= fa-xIf fx is symmetric about x=b thenfb+x= fb-xWe havefa+x= fa-xfb+x= fb-xTo prove that fx is periodic, we need to show fx+T=fx, for some t≠0fa+x= fa-xReplace x by x-afa+x-a= fa-x-afx=f2a-x ____________1fb+x= fb-xReplace x by x-bfb+x-b= fb-x-bfx=f2b-x ____________2From equation 1 and 2f2b-x=f2a-xreplace x by -xf2b+x=f2a+xfx+2a=fx+2bReplace x by x-2afx-2a+2a=fx-2a+2bfx=fx-2a+2bfx=fx+2b-aHence we have shown that fx=fx+T, where T=2b-afx is periodic

Q). If f(x) is symmetrical about the line x = a, x = b (a > b). Prove that f(x) is periodic.