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The problem asks to find the equation of the image (bayangan) of a linear function \( f(x) = 3x + 6 \) after it undergoes a dilation transformation with the center at \( (0,0) \) and a scaling factor of 2. Additionally, it asks to determine the x-intercept of the resulting function after the transformation.

### Step 1: Understanding the Dilation Transformation
A dilation with a scale factor \( k \) and center at \( (0,0) \) affects both the x and y coordinates of every point on the graph of the function by multiplying them by \( k \). For a linear function, this transformation scales both the slope and the y-intercept.

Given that the scale factor is 2, the function will transform as follows:
- The slope will be scaled by 2.
- The y-intercept will also be scaled by 2.

### Step 2: Applying the Dilation
The original function is:

\[
f(x) = 3x + 6
\]

After dilation by a factor of 2, the new function will be:

\[
f'(x) = 2 \cdot (3x + 6)
\]

Simplifying:

\[
f'(x) = 6x + 12
\]

### Step 3: Finding the x-Intercept
The x-intercept occurs when \( f'(x) = 0 \). To find this:

\[
6x + 12 = 0
\]

Solving for \( x \):

\[
6x = -12
\]
\[
x = -2
\]

Thus, the x-intercept of the new function is at \( x = -2 \).

### Final Answer:
- The equation of the image after dilation is \( f'(x) = 6x + 12 \).
- The x-intercept of this function is \( x = -2 \).

Would you like more details or further explanation?

Here are some related questions:
1. What happens to the slope and intercept of a function after dilation with a different center than \( (0,0) \)?
2. How do other transformations like translations affect linear functions?
3. Can you describe the effect of a dilation with a factor less than 1 on the graph of a function?
4. How does a reflection along the x-axis affect the equation of a linear function?
5. How can we graphically represent dilation on a coordinate plane?

**Tip:** When scaling a linear function, the slope and intercept are scaled by the same factor, but the nature of the line (whether it crosses the axes) depends on this factor as well.

Tentukan persamaan bayangan dari fungsi linear f(x) = 3x + 6 jika didilatasikan dengan pusat (0,0) dan faktor skala 2, kemudian tentukan titik potong fungsi hasil peta dengan sumbu X.

Learn how to find the image of the linear function f(x) = 3x + 6 after a dilation centered at (0, 0) with a scale factor of 2. This step-by-step solution includes how to determine the new function's x-intercept. Suitable for high school math students.

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