A function f(x) is transformed to 2f[-3(x+18)]+8  .
Let (6,9) be a point on the transformed graph.
What is x-coordinate of the corresponding  pre-image point?

Given that \((6, 9)\) is a point on the transformed graph of the function \( f(x) \) transformed to \( 2f[-3(x+18)]+8 \), we need to find the x-coordinate of the corresponding pre-image point on the original function \( f(x) \).

The transformation applied to \( f(x) \) is \( 2f[-3(x+18)]+8 \). We can break this transformation down step-by-step:

1. **Horizontal transformation**:
   - \( f(x) \) is transformed to \( f(x+18) \) which shifts the graph 18 units to the left.
   - Next, \( f(x+18) \) is transformed to \( f(-3(x+18)) \) which reflects the graph across the y-axis and then compresses it horizontally by a factor of 3.

2. **Vertical transformation**:
   - The function \( f(x) \) is multiplied by 2, resulting in \( 2f(-3(x+18)) \).
   - Finally, 8 is added to the function, giving us \( 2f(-3(x+18)) + 8 \).

Given that the point \( (6, 9) \) is on the transformed graph, this implies:
\[
2f[-3(6 + 18)] + 8 = 9
\]

First, simplify inside the argument of \( f \):
\[
-3(6 + 18) = -3 \times 24 = -72
\]

Thus, the equation becomes:
\[
2f(-72) + 8 = 9
\]

Subtract 8 from both sides:
\[
2f(-72) = 1
\]

Divide both sides by 2:
\[
f(-72) = \frac{1}{2}
\]

The x-coordinate \(-72\) is the pre-image point on the function \( f(x) \) before it was transformed. Hence, the corresponding pre-image x-coordinate is:

\[
x = -72
\]

Would you like more details or have any questions about the transformation process?

Here are some questions you might want to ask next:
1. How do horizontal and vertical transformations affect the graph of a function?
2. Can you explain the impact of reflecting a function across the y-axis?
3. How do you handle multiple transformations applied to a function?
4. What is the difference between horizontal compression and horizontal stretch?
5. Can you provide another example of finding the pre-image of a point after transformations?

**Tip:** When working with transformations, always apply horizontal transformations (inside the function) before vertical transformations (outside the function) for a clear and accurate process.

Learn how to find the pre-image point of a function transformation using a detailed step-by-step guide. This problem involves function transformations, reflections across the y-axis, and horizontal compressions. Suitable for high school students studying function transformations in algebra.

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