To make a rigid motion, e have to maintain the relative positions and distances of the points. To accomplish this, we can't multiply f(x) or x alone.
In the first option, only f(x) is mutiplied, which makes it a non rigid motion.
In the third option, both are multiplied, by this don't garantee a rigid motion depending on the function.
I the fourth option, only x is multiplied, not a ridig motion.
In the second option, there is a substraction in x, which moves all points equally in x axis, and a sum in the f, which moves all the points equally in the y axis. So this is the right option.
Concluding, the option that transforms f(x) in a ridig motion is the second:
