$\"\"$

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Consider $\"\"$ and $\"\"$ are the even functions.

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From the definitions $\"\"$ and $\"\"$ .

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Find the addition $\"\"$ .

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$\"\"$

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$\"\"$ .

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If $\"\"$ and $\"\"$ are the even functions, then the addition $\"\"$ is an even function.

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$\"\"$

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Consider $\"\"$ and $\"\"$ are the odd functions.

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From the definitions $\"\"$ and $\"\"$ .

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Find the addition $\"\"$ .

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$\"\"$

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$\"\"$ .

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If $\"\"$ and $\"\"$ are the odd functions, then the addition $\"\"$ is an odd function.

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$\"\"$

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Consider $\"\"$ is an even function and $\"\"$ is an odd function.

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From the definitions $\"\"$ and $\"\"$ .

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Find the addition $\"\"$ .

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$\"\"$

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This is not equal to either $\"\"$ or $\"\"$ .

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Thus, $\"\"$ is neither even nor odd function.

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$\"\"$

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If $\"\"$ and $\"\"$ are the even functions, then the addition $\"\"$ is an even function.

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If $\"\"$ and $\"\"$ are the odd functions, then the addition $\"\"$ is an odd function.

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If one of the function is even and other is odd, then the addition $\"\"$ is neither even nor odd function.