$\"\"$

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

From the definitions $\"f(-x)=f(x)\"$ and $\"g(-x)=g(x)\"$ .

Find the addition $\"f+g\"$ .

$\"(f+g)(-x)=f(-x)+g(-x)\"$

$\"(f+g)(-x)=f(x)+g(x)\"$

$\"(f+g)(-x)=(f+g)(x)\"$ .

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

$\"\"$

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

From the definitions $\"f(-x)=-f(x)\"$ and $\"g(-x)=-g(x)\"$ .

Find the addition $\"f+g\"$ .

$\"(f+g)(-x)=f(-x)+g(-x)\"$

$\"(f+g)(-x)=\\left$

$\"(f+g)(-x)=-\\left$

$\"(f+g)(-x)=-(f+g)(x)\"$ .

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

$\"\"$

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

From the definitions $\"f(-x)=f(x)\"$ and $\"g(-x)=g(x)\"$ .

Find the addition $\"f+g\"$ .

$\"(f+g)(-x)=f(-x)+g(-x)\"$

$\"(f+g)(-x)=f(x)+(-g(x))\"$

$\"(f+g)(-x)=f(x)-g(x)\"$

This is not equal to either $\"(f+g)(x)\"$ or $\"-(f+g)(x)\"$ .

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

$\"\"$

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

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

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