The graph of $$ f(x)=|x| $$ is transformed to $$ g(x)=|x+1|-7 $$. On which interval is the function decreasing? $$ (-\infty,-7) $$ $$ (-\infty,-1) $$ $$ (-\infty, 1) $$ $$ (-\infty, 7) $$

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To determine where the function $$ g(x) = |x + 1| - 7 $$ is decreasing, let's analyze its structure:

1. **Understanding the Transformation:**
   - The base function is $$ f(x) = |x| $$, which is a V-shaped graph with its vertex at $$ (0, 0) $$.
   - The transformed function $$ g(x) = |x + 1| - 7 $$ shifts the graph horizontally and vertically:
     - **Horizontal Shift:** $$ |x + 1| $$ shifts the graph 1 unit to the **left**, placing the vertex at $$ x = -1 $$.
     - **Vertical Shift:** Subtracting 7 moves the entire graph **down** by 7 units.

2. **Analyzing the Vertex:**
   - The vertex of $$ g(x) $$ is at $$ x = -1 $$.
   - For absolute value functions of the form $$ |x + h| + k $$:
     - The function **decreases** as $$ x $$ moves **toward** the vertex from the left.
     - The function **increases** as $$ x $$ moves **away** from the vertex to the right.

3. **Determining the Decreasing Interval:**
   - Since the vertex is at $$ x = -1 $$, $$ g(x) $$ will **decrease** for all $$ x $$ values **less than -1**.
   - Therefore, the function is decreasing on the interval $$ (-\infty, -1) $$.

**Conclusion:**
The function $$ g(x) = |x + 1| - 7 $$ is decreasing on the interval $$ (-\infty, -1) $$.

**Answer:** $$ (-\infty,-1) $$
The function $$ g(x) = |x + 1| - 7 $$ is decreasing on the interval $$ (-\infty, -1) $$.

The graph of is transformed to . On which interval is the function decreasing?

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