Final answer:
Multiplying function f(x) by scalar c to get f(cx) alters the form of the function by changing inputs instead of scaling outputs, thus breaking the rule of scalar multiplication in vector spaces.
Step-by-step explanation:
The question pertains to whether the functions f(x) = x² and g(x) = 5x, treated as vectors in the vector space of all real functions, satisfy the standard rules for scalar multiplication. If we multiply a function f by a scalar c, standard vector space rules dictate that we should multiply the output by c, not input. That is, c⋅f(x) = c⋅(x²) and not f(cx) = (cx)². In this context, if multiplying f(x) by c gives the function f(cx), this breaks the rule that scalar multiplication should alter the magnitude of outputs, not change the form of the function itself by altering inputs.
It is important to differentiate between the two types of multiplication in vector spaces: scalar multiplication affecting the magnitude of the vector (function in this case), and the vector product, which relates to direction. Scalar multiplication should maintain the functional form and only scale outputs, and it must obey properties such as distribution and association similar to the multiplication of real numbers.
When a function as a vector is multiplied by a scalar, the expected result would be every y-value of the function gets scaled by that scalar, keeping the x-values constant. Therefore, the multiplication rule that is broken when f(x) is multiplied by scalar c to get f(cx) is the rule of scaling output values and not input values.