If f(x) and g(x) are odd functions, show that the composite function f(g(x)) is also odd.

AdamTaurus
Oct 10, 2017

#2**+3 **

This is not likely to be a very good 'proof' but here goes.

since f(x) and g(x) are odd funtion,

\(f(-x)=-f(x)\) and \(g(-x)=-g(x)\)

Let \(x_1\) be a point such that

\(a=g(x_1) \qquad -a=g(-x_1)\)

\(f(g(x_1))=f(a) \qquad f(g(-x_1))=f(-g(x_1))=f(-a)=-f(a)\\ so\\ f(-g(x_1))=-f(g(x_1)) \\ \text{by definition this means that }f(g(x)) \text{ is an odd function.}\)

Melody
Oct 10, 2017