Even function+even function = even function
Odd Function * Odd Function = Even Function
Even Function * Even Function = Even Function
Odd Function * Even Function = Odd Function
Parity of compound function: even if it is internal, it is odd if it is external;
Monotonicity of compound function: mutual increase and mutual decrease.
Extended data:
Prove by definition
1, odd function plus odd function equals odd function.
Let f(x) and g(x) be odd function, and h(x)=f(x)+g(x).
Then h (-x) = f (-x)+g (-x) =-f (x)-g (x) =-(f (x)+g (x)) =-h (x).
So h(x) is odd function.
2. Even function plus even function equals even function.
Let f(x) and g(x) be even functions, and h(x)=f(x)+g(x).
Then h (-x) = f (-x)+g (-x) = f (x)+g (x) = h (x).
So h(x) is an even function.
3. odd function addition function is equal to nonsingular non-even function.
Let f(x) be odd function, g(x) be even function, and h(x)=f(x)+g(x).
Then h(-x)=f(-x)+g(-x)=-f(x)+g(x)
Obviously, h(-x) is not equal to h(x), nor is it equal to -h(x).
Therefore, h(x) is a parity function.
4. Constant terms are regarded as even functions.
Let f(x)=k(k is a constant)
f(-x)=k=f(x)
So f(x) is an even function.