To determine the interval on which the function [tex]\( g(x) = |x+1| - 7 \)[/tex] is decreasing, let's analyze the transformations applied to the original function [tex]\( f(x) = |x| \)[/tex].
1. Understanding the Absolute Value Function:
The function [tex]\( f(x) = |x| \)[/tex] has a V-shape, with the vertex at [tex]\( x = 0 \)[/tex]. It decreases on the interval [tex]\( (-\infty, 0) \)[/tex] and increases on the interval [tex]\( (0, \infty) \)[/tex].
2. Transformation [tex]\( g(x) = |x+1| - 7 \)[/tex]:
- [tex]\( |x+1| \)[/tex] moves the graph of [tex]\( |x| \)[/tex] to the left by 1 unit. So, the vertex of the absolute value part is now at [tex]\( x = -1 \)[/tex].
- Subtracting 7 shifts the entire graph downward by 7 units, but this vertical shift does not affect the intervals where the function is increasing or decreasing.
3. Vertex of the Transformed Function [tex]\( g(x) \)[/tex]:
The vertex of [tex]\( g(x) = |x+1| - 7 \)[/tex] is at [tex]\( x = -1 \)[/tex]. This is because the transformation [tex]\( |x+1| \)[/tex] changes the location of the minimum point of the absolute value function to [tex]\( x = -1 \)[/tex].
4. Intervals of Decrease:
The function [tex]\( g(x) = |x+1| - 7 \)[/tex] will follow the behavior of the absolute value function [tex]\( |x+1| \)[/tex]. Therefore:
- It will decrease on the interval to the left of the vertex, which is [tex]\( (-\infty, -1) \)[/tex].
- It will increase on the interval to the right of the vertex, which is [tex]\( (-1, \infty) \)[/tex].
From this analysis, the function [tex]\( g(x) \)[/tex] is decreasing on the interval [tex]\( (-\infty, -1) \)[/tex].
Thus, the correct interval is:
[tex]\[ (-\infty, -1) \][/tex]
