f(x) = c - x is not written as f(x) = c - mx because the problem you are given has explicitly told you that the value of m would be 1 in your version of the equation.
I think that the larger problem here is that you might not be seeing this:
f(x) = c - x is the same as f(x) = -x + c because the commutative law allows you to rearrange the order of two items being added. Maybe your textbook wanted to save on the price of printing a '+' sign?
This is a line with a y-intercept value of c and a slope of -1 as the coefficient of x is -1.
A family of functions means that there is a bunch of functions that are all the same except they can differ by a constant. The c term in the equation is the constant. Graphically, this means that the functions behave the same way, they are just shifted up or down from one another.
j(x) = 3x + 3, k(x) = 3x - 2 belong to the same family of functions because they are the same function with just a different constant.