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.