To determine the interval on which the function [tex]\( g(x) = |x + 1| - 7 \)[/tex] is decreasing, we need to carefully analyze the transformations made to the base function [tex]\( f(x) = |x| \)[/tex].
1. Understanding the Base Function [tex]\( f(x) = |x| \)[/tex]:
- The function [tex]\( f(x) = |x| \)[/tex] represents the absolute value of [tex]\( x \)[/tex].
- The graph of [tex]\( f(x) = |x| \)[/tex] consists of two linear segments:
- [tex]\( f(x) = -x \)[/tex] for [tex]\( x < 0 \)[/tex]
- [tex]\( f(x) = x \)[/tex] for [tex]\( x \ge 0 \)[/tex]
2. Transformations Applied to the Base Function:
- The function [tex]\( g(x) = |x + 1| - 7 \)[/tex] involves two transformations:
- Horizontal Shift: The term [tex]\( |x + 1| \)[/tex] represents a horizontal shift of 1 unit to the left of the graph of [tex]\( f(x) = |x| \)[/tex]. This means that the vertex of [tex]\( |x| \)[/tex], which is at [tex]\( x = 0 \)[/tex], is now at [tex]\( x = -1 \)[/tex].
- Vertical Shift: The term [tex]\( -7 \)[/tex] represents a downward shift of 7 units. The entire graph of the function is moved 7 units downward.
3. Analyzing the Decreasing Interval:
- For the base function [tex]\( f(x) = |x| \)[/tex], it is decreasing in the interval [tex]\( (-\infty, 0) \)[/tex].
- With the horizontal shift by 1 unit to the left, the decreasing part of the function [tex]\( |x| \)[/tex] (i.e., [tex]\( f(x) = -x \)[/tex]) is now in the interval [tex]\( (-\infty, -1) \)[/tex].
4. Verifying the Decreasing Interval for [tex]\( g(x) \)[/tex]:
- The vertical shift does not affect the interval where the function is decreasing; it only affects the vertical position of the graph.
- Therefore, [tex]\( g(x) \)[/tex] will be decreasing in the interval [tex]\( (-\infty, -1) \)[/tex].
In conclusion, the interval on which the function [tex]\( g(x) = |x + 1| - 7 \)[/tex] is decreasing is:
[tex]\[ (-\infty, -1) \][/tex]
Hence, the correct answer is:
[tex]\[
(-\infty, -1)
\][/tex]
