To check: Whether is always an odd function if g is odd; also check what would be the result if f is an odd function and if f is even function.
The composition function is not always an odd function.
Iff is an odd function, then the composition function is always an odd function.
If f is an even function, then the composition function is always an even function.
The function is an odd function.
The composite function is,
That is .
The composition function is said to be an odd function if , is said to be an even function if .
So, first obtain the value of as shown below.
Here, it cannot be concluded that whether the composition function is an even function or an odd function as it depends on the nature of the function . Thus, is not always an odd function.
There exist two cases which are explained as follows.
If is an odd function, then the equation (1) becomes,
Thus, the composite function is an odd function when the functions and are odd functions.
If is an even function, then the equation (1) becomes,
Thus, the composite function is an even function when the function is even and the function is odd functions.
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