Re: Distributive property of functions

On Mon, 31 Mar 2008 07:19:47 -0700 (PDT), Olumide <50295@xxxxxx> fed
this fish to the penguins:

Hello -

I hope this isn't too trivial to ask but, I'm working through a proof
that appears to rely on the distributive property of polynomials, i.e.

(f + g)(x) = f(x) + g(x)

without saying saying so. I've googled a bit, and I've found that
trigononometric functions e.g. sin(x) do not have this property,
although polynomials appear to (I've done a simple numerical example
in which g = x**3 and f = x**2 ). I guess my question then is: what
sort of functions f(x) and g(x) have this property?


I think there is a confusion on your part here: The equality

(f + g)(x) = f(x) + g(x)

is true for every pair of functions (into the reals say) by
*definition*, in other words, that is how we *define* the (pointwise)
sum f + g of two functions f and g. Nothing mysterious here. On the
other hand your comment that the trigonometric functions do not have
this property *is* mysterious. Perhaps you meant *linearity*, e.g.
indeed

sin(x + y) = sin(x) + sin(y)

is false, but that is very different from what you wrote.

Hope it helps, regards,
G. Rodrigues
.