Final answer:
The rule that is broken when multiplying f(x) by a constant c and getting f(cx) is the scalar multiplication of functions. Multiplying a function by a constant should vertically stretch or compress the graph, not change the input variable x. The rules of mathematics like addition and multiplication of fractions are universally valid and independent of the order or context.
Step-by-step explanation:
The student's question pertains to multiplication rules and constants within functions. When you multiply a function, f(x), by a constant, c, the expected result is cf(x), not f(cx). This is because, mathematically, c acts as a scalar that stretches or compresses the graph of f(x) vertically and does not affect the input value x.
Multiplying a function by a variable or another function, such as f(x) multiplied by λ, implies a direct variation where the product is a constant. As stated in the discussions, the smaller one factor is, the larger the other must be to maintain the constant product. This concept is reflective of the inverse variation or inverse proportion. Discussing the general multiplication of fractions, the rule symbolically can be expressed as (a/b) × (c/d) = (ac)/(bd) where a, b, c, and d are numbers, and b and d are not zero. This rule allows us to multiply any two fractions by multiplying their numerators together and their denominators together separately.
The commutative property of addition, such as A + B = B + A, demonstrates the flexibility of order in addition, as with the example 2 + 3 or 3 + 2, yielding the same result. This property does not apply to functions under multiplication unless explicitly defined. The rules of mathematics, such as 12 + 19 = 31, are universal truths that hold irrespective of geographical location or cultural context. An error in basic arithmetic, like adding 12 and 19 and getting 32, is a case of not applying the correct rule of addition, regardless of whether the person is an instructor or a peasant.