Why does my book say that "since a major change takes place at (1,1), the expression in the absolute value should equal to zero at x=1?

Question

Question I am referring to:

Why does the absolute value portion of the expression of the function this graph corresponds to has to be 0 when x=1?

Here are the five choices:

Answer 1

Answer B is correct. You know that the function is a sum of two different functions, a linear function and an absolute value function.

The absolute value function $f(x) = |g(x)|$ has a major change when $f(x) = 0$. But in this graph, there is a major change at $x = 1$, so $f(x) = 0$ at $x = 1$, and the absolute value function must be translated by one unit to the right to make the function work. In other words, $g(x) = x - 1$ (translated one unit to the right), and the answer with the $f(x) = |x-1|$ term is correct.