Composite of odd functions is odd

Statement

Statement for two functions

Suppose $f$ and $g$ are odd functions so that the composite $f \circ g$ makes sense. Then, $f \circ g$ is also an odd function.

Note that composition of functions does not commute, so if we can make sense of both $f \circ g$ and $g \circ f$, these are both (possibly equal, possibly distinct) odd functions.

Statement for more than two functions

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Related facts

Similar facts

• Odd functions form a vector space
• Inverse function of odd function is odd

Similar facts for even functions

• Composite of even function with odd function is even
• Composite of any function with even function is even