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# Composite Functions

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

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#1
+7048
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I want to know how to do this problem too.....

hectictar  Oct 10, 2017
#2
+92439
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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