Final answer:
The student is looking for functions from Q to R that are additive, meaning f(x+y) = f(x) + f(y) for all x, y in Q. These functions are linear and the additive property shown is akin to commutativity in number addition. The general form is f(x) = cx for some constant c in R, but only rational arguments are considered.
Step-by-step explanation:
The student's question seeks to find all functions f from the rational numbers (Q) to the real numbers (R) that satisfy the functional equation f(x+y) = f(x) + f(y) for all x and y in Q. This property is known as additivity or being Cauchy's functional equation. Such functions are linear over the rationals, implying that f(qx) = qf(x) for any rational number q and x in Q. Additionally, the additive property is known as commutativity, where A+B=B+A, which holds for the addition of rational numbers.
For instance, if you choose a rational number r, then f(r) could be any real number since f(0) = f(r-r) = f(r) + f(-r), leading to f(0) + f(-r) = f(r). Assuming that f(0) = 0, we get f(-r) = -f(r), which is coherent with the property of additive inverses. However, it is important to note that while these functions are linear over the rationals and hence can be represented as f(x) = cx for some constant c in R, this does not necessarily extend to real numbers due to the requirement that x and y must be rational.