What are the transformations from g(x) = -3, f(x + 6) - 8 to f(x)?

To transform \( g(x) = -3f(x + 6) - 8 \) into \( f(x) \), you need to reverse the transformations applied to the function step by step. Let's break it down:

### Transformations:
1. **Horizontal Shift (inside the function)**:
   - \( f(x + 6) \) indicates a **horizontal shift** to the **left** by 6 units. This shift moves the graph to the left.

2. **Vertical Stretch and Reflection (multiplied by -3)**:
   - \( -3f(x + 6) \) means the function is vertically **stretched** by a factor of 3 and then **reflected** across the x-axis due to the negative sign. This changes the shape of the graph.

3. **Vertical Shift (outside the function)**:
   - The subtraction of 8, \( -3f(x + 6) - 8 \), indicates a **vertical shift** **downward** by 8 units. This shifts the entire graph downward by 8.

### In reverse, to go back from \( g(x) = -3f(x + 6) - 8 \) to \( f(x) \):
1. **Undo the vertical shift**: Add 8 to both sides to undo the downward shift.
   \[
   g(x) + 8 = -3f(x + 6)
   \]

2. **Undo the vertical stretch and reflection**: Divide both sides by -3 to reverse the stretch and reflection.
   \[
   \frac{g(x) + 8}{-3} = f(x + 6)
   \]

3. **Undo the horizontal shift**: Replace \( x + 6 \) with \( x \) to reverse the horizontal shift.
   \[
   f(x) = \frac{g(x - 6) + 8}{-3}
   \]

This gives you the original function \( f(x) \).

### Final Expression:
\[
f(x) = \frac{g(x - 6) + 8}{-3}
\]

Would you like further clarification or details on any of these steps?

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Here are 5 related questions for further exploration:
1. How do horizontal and vertical shifts affect the graph of a function?
2. What is the effect of reflecting a function across the x-axis?
3. How do you identify transformations from the equation of a function?
4. What are the differences between a vertical stretch and a vertical compression?
5. How do translations differ from dilations in function transformations?

**Tip:** Always apply transformations in the correct order: start with horizontal shifts, followed by stretches/reflections, and finally vertical shifts.

What are the transformations from g(x) = -3f(x + 6) - 8 to f(x)?

Discover the step-by-step process of transforming the function g(x) = -3f(x + 6) - 8 back to its original form f(x). This solution covers horizontal and vertical shifts, reflection, and vertical stretching, ideal for students in Grades 10-12 studying function transformations.

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