Question

Suppose we are given two functions, say r(x) and s(x).
a) Is it true that r(x) + s(x) = s(x) + r(x) for all values of x? Why or why not?
b) Is it true that r(x) − s(x) = s(x) − r(x) for all values of x? Why or why not?

c) Is it true that r(0) = 0? Why or why not?

Answer 1

Since r(x) and s(x) are assumed to be real functions, r(x) + s(x) = s(x) + r(x) by the commutative property of addition.

Subtraction, however, does not have the same commutative property. Note that s(x) - r(x) = -[r(x) -s(x)]. Therefore the question is basically asking
r(x) - s(x) = -[r(x) -s(x)] ??
Unless these functions are each equivalent to 0 (maybe theres another case), this is certainly not true.

We do not even know what r(x) is. r(x) is arbitrary and may be something like x + 5. Therefore we cannot say r(0) = 0

Answer 2

okay, thanks so much!