2. Given the graph of $$ f(x)=\sqrt{x} $$, what happens to the graph when $$ f(x) $$ is replaced by $$ 2 f(-(x+4))-3 $$ ?

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To understand the transformation of the graph of $$ f(x) = \sqrt{x} $$ into $$ F(x) = 2 f(-(x+4)) - 3 $$, let's break down each step of the transformation:

1. **Horizontal Shift:**
   - **Inside the function:** $$ -(x + 4) $$ indicates a horizontal shift.
   - **Interpretation:** 
     - The $$ +4 $$ inside the function shifts the graph **4 units to the left**.
     - The negative sign $$ - $$ reflects the graph across the **y-axis**.

2. **Vertical Stretch:**
   - **Outside the function:** The coefficient $$ 2 $$ multiplies the entire function.
   - **Interpretation:** This **stretches the graph vertically by a factor of 2**. Every y-value of the original function is doubled.

3. **Vertical Shift:**
   - **Outside the function:** The $$ -3 $$ subtracts from the entire function.
   - **Interpretation:** This **shifts the graph **3 units downward**.

### Summary of Transformations:
1. **Shift 4 units to the left:** $$ x $$ replaced by $$ x + 4 $$.
2. **Reflect across the y-axis:** $$ x + 4 $$ replaced by $$ -(x + 4) $$.
3. **Vertical stretch by a factor of 2:** $$ \sqrt{-(x + 4)} $$ multiplied by 2.
4. **Shift downward by 3 units:** Subtract 3 from the entire function.

### Visual Representation:
- **Original Function:** Starts at the origin (0,0) and curves upward to the right.
- **Transformed Function:**
  - Starts at (-4, -3) due to the horizontal shift and vertical shift.
  - Reflected across the y-axis, so it now curves to the left.
  - Stretched vertically, making it steeper.

### Final Transformed Function:
$$ F(x) = 2\sqrt{-(x + 4)} - 3 $$

**Graphically, the transformed $$ F(x) $$ is obtained by shifting $$ f(x) = \sqrt{x} $$ left 4 units, reflecting it across the y‑axis, stretching it vertically by a factor of two, and then moving it down three units.**
The graph of $$ f(x) = \sqrt{x} $$ is transformed by shifting it 4 units to the left, reflecting it across the y-axis, stretching it vertically by a factor of 2, and then moving it down 3 units. The final transformed function is $$ F(x) = 2\sqrt{-(x + 4)} - 3 $$.

2. Given the graph of \( f(x)=\sqrt{x} \), what happens to the graph when \( f(x) \) is replaced by \( 2 f(-(x+4))-3 \) ?

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