Let's analyze the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to determine the translation applied.
Given:
- [tex]\( f(x) = x \)[/tex]
- [tex]\( g(x) = |x| + 3 \)[/tex]
We need to find the relationship and transformation between these two functions.
1. Starting with [tex]\( f(x) = x \)[/tex]:
- This is a linear function with a slope of 1, passing through the origin (0,0).
2. Looking at [tex]\( g(x) = |x| + 3 \)[/tex]:
- The absolute value function [tex]\( |x| \)[/tex] creates a V-shaped graph that meets at the point (0,0) and opens upwards.
- Adding 3 to [tex]\( |x| \)[/tex] shifts the entire graph of [tex]\( |x| \)[/tex] vertically upward by 3 units.
To summarize, the function [tex]\( g(x) = |x| + 3 \)[/tex] results from a vertical translation of the graph of [tex]\( |x| \)[/tex] by 3 units in the positive direction. We can think of the absolute value operation [tex]\( |x| \)[/tex] as a part of the transformation that changes the shape from a line to a V-shape, but the +3 clearly indicates a vertical shift upward.
Therefore, the correct answer is:
a vertical translation of 3 units upward
