Prove that (f+g)(x) is an odd function, if f and g are odd functions (Stewart, Calculus)

Suppose f(x) and g(x) are odd functions. Prove that (f+g)(x) is also an odd function. 

Answer: 

1. Strategy

By definition, f is an odd function if and only if f(-x) = - f(x)

To show (f+g) is an odd function, we need to show (f+g)(-x) = - (f+g)(x)

2. Explanation

Since $f(x)$ and $g(x)$ are odd functions

$\Rightarrow f(-x) =-f(x)$ and $g(-x) =-g(x)$

By definition of sum of functions.

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

$=-f(x)-g(x)$

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

$=-(f+g)(x)$ (by definition of sum of functions)

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

$\Rightarrow f+g$ is an odd function. Q.E.D.