To determine the direction in which the graph of [tex]\( f(x) = |x| \)[/tex] must be shifted to produce the graph of [tex]\( g(x) = |x-3| \)[/tex], we need to understand how transformations affect the graph of a function.
The function [tex]\( f(x) = |x| \)[/tex] is a standard absolute value function. When a constant is subtracted inside the absolute value, as in [tex]\( g(x) = |x-3| \)[/tex], it causes a horizontal shift. Specifically:
- When you have [tex]\( f(x - c) \)[/tex], the graph shifts [tex]\( c \)[/tex] units to the right if [tex]\( c \)[/tex] is positive.
- When you have [tex]\( f(x + c) \)[/tex], the graph shifts [tex]\( c \)[/tex] units to the left if [tex]\( c \)[/tex] is positive.
In the function [tex]\( g(x) = |x-3| \)[/tex], the [tex]\( -3 \)[/tex] inside the absolute value indicates that the graph will shift 3 units to the right.
Therefore, the graph of [tex]\( f(x) = |x| \)[/tex] must be shifted to the right to produce the graph of [tex]\( g(x) = |x-3| \)[/tex].
The correct answer is:
D. right
