Compare and contrast the results of adding or multiplying a value outside the argument of a function to the result of adding or multiplying a value inside the argument of a function.

Only in special funcitons, specifically linear functions whose graphs pass through the origin, it is the same to put the constant inside the argument or outside

Step-by-step explanation:

If the function has the form f (x) = a*x, where a is a constant, then we have that f (c*x) = a * (c*x) = c * (a*x) = c*f (x). This kind of functions are proportional to the identity function f (x) = x, and they comprehend the linear functions whose graph pass through the origin.

For other linear function this property isnt true. For example if f (x) = x+4, then

f (4) = 4+4 = 8,

f (2*4) = 8+4 = 12

2*f (4) = 2*8 = 16

Thus, 2*f (4) is not f (2*4).

This property also isnt true for quadratic functions for example. If f (x) = x², then f (1) = 1, thus 3*f (1) = 3, however, f (3*1) = 3² = 9.

There might be coincidences for specific values, for example if f (x) = (x-1) * (x-2), then f (2*1) = 2*f (1) = 0, however, for any other constant the result is not the same (for example f (0*1) = (-1) * (-2) = 2, and 0*f (1) = 0).

If we want a function to satisfy the property f (c*x) = c*f (x) for any c, x, then it should be true that f (c) = f (c*1) = c*f (1). This means that if f (1) = a, then f (c) = c*a, so, in other words, f (x) = a*x = f (1) * x.