The shape of $$ f\left(x\right) $$= $$  $$$$ \sqrt{x} $$, but shifted six units to the left and then reflected in both the x-axis and y-axis

To transform the function $ f(x) = \sqrt{x} $ according to the given operations, we'll proceed step by step:

### 1. **Shifting Six Units to the Left:**
   - When a function is shifted to the left by $ c $ units, you replace $ x $ with $ x + c $ in the function.
   - So, shifting $ f(x) = \sqrt{x} $ six units to the left results in:
     \[
     f(x) = \sqrt{x + 6}
     \]

### 2. **Reflecting in the x-axis:**
   - Reflecting a function across the x-axis involves multiplying the function by $-1$.
   - Applying this reflection to $ f(x) = \sqrt{x + 6} $ gives:
     \[
     f(x) = -\sqrt{x + 6}
     \]

### 3. **Reflecting in the y-axis:**
   - Reflecting a function across the y-axis involves replacing $ x $ with $-x$ in the function.
   - Applying this reflection to $ f(x) = -\sqrt{x + 6} $ gives:
     \[
     f(x) = -\sqrt{-x + 6}
     \]
   - This can also be written as:
     \[
     f(x) = -\sqrt{6 - x}
     \]

### Final Answer:
The function $ f(x) = \sqrt{x} $, after being shifted six units to the left and reflected in both the x-axis and y-axis, becomes:
\[
f(x) = -\sqrt{6 - x}
\]

Would you like a detailed explanation of any of these steps or more examples?

Here are 8 related questions to expand on this topic:
1. What happens to the domain and range of $ \sqrt{x} $ after these transformations?
2. How does reflecting a function in the y-axis differ from reflecting in the x-axis?
3. What is the graph of the original function $ \sqrt{x} $ and how do these transformations affect it?
4. What other transformations can be applied to functions like $ \sqrt{x} $?
5. Can you provide the transformation for shifting a function 4 units up and reflecting it across the x-axis?
6. How would you find the inverse of the transformed function $ -\sqrt{6 - x} $?
7. How does the vertex of a parabolic function shift under similar transformations?
8. What are the key properties of square root functions and how are they altered by transformations?

**Tip:** Understanding how basic transformations (shifts, reflections, stretches) affect the graph of a function will help you visualize complex functions more easily.

Explore how the function $ f(x) = \sqrt{x} $ transforms when shifted six units to the left and reflected in both the x-axis and y-axis. Understand the step-by-step process of these transformations with clear examples and explanations. Suitable for advanced high school students and anyone interested in mathematical functions and transformations.

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