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

AdamTaurus  Oct 10, 2017
 #1
avatar+7266 
+2

I want to know how to do this problem too.....

hectictar  Oct 10, 2017
 #2
avatar+93325 
+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

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