Where in the world did "= 0" come from?

g(x)= (1/2)[f(x)+ f(-x)] so g(-x)= (1/2)[f(-x)+ f(x)] and, since addition is commutative, that is the same as g(-x)= (1/2)[f(x)+ f(-x)]. What does that tell you?

Now, how did f(-x)- f(x) become "-2f(x)"?? It is NOT given that f(-x)= -f(x). That is, you are NOT told that f(x) is itself an odd function.

h(x)= (1/2)[f(x)- f(-x)] so h(-x)= (1/2)[f(-x)- f(x)]= (1/2)[-(f(x)- f(-x))]= -(1/2)[f(x)- f(-x)].

What is g(x)+ h(x)?

These are, by the way, refered to as the even and odd "parts" of f(x).

For example, if

then

and

. cosh(x) and sinh(x)

**are** the "even and odd parts" of