Final answer:
The statement about function additivity is false because it only applies to specific functions, particularly linear ones. Non-linear functions do not generally satisfy this property as demonstrated with the function f(x) = x^2. Hence, not every function can be assumed to have the additive property just like ordinary number addition.
Step-by-step explanation:
The statement that if f is a function, then f(s + t) = f(s) + f(t) is not necessarily true for all functions. It describes a specific property known as additivity or being an additive function. This property holds true for linear functions, where the output is directly proportional to the input. However, not all functions exhibit this property. The given rule only holds for functions where the principle of superposition applies.
For example, consider the linear function f(x) = 2x. For this function, f(s + t) = 2(s + t) = 2s + 2t = f(s) + f(t), which confirms that the statement is true. However, if we consider a non-linear function like f(x) = x2, then f(s + t) = (s + t)2 does not equal f(s) + f(t) = s2 + t2. Thus, the original statement is false for the non-linear function.
In addition, the commutative property of addition, stated as A+B=B+A, is true for the addition of ordinary numbers. This means you get the same result whether you add 2 + 3 or 3 + 2, which further illustrates that additivity is a distinct concept from commutativity.