Given that f(x) = root x, the graph of function g can be produced as by transforming f as follows: Shifting f to the right 7 units, vertically stretching the result by a factor of 3. What is the formula for g?

To find the formula for the function \( g(x) \), we start with the function \( f(x) = \sqrt{x} \) and apply the given transformations:

1. **Shifting \( f(x) \) to the right by 7 units**:
   - To shift a function to the right by \( h \) units, you replace \( x \) with \( x - h \).
   - Therefore, the function after shifting to the right by 7 units is \( f(x - 7) = \sqrt{x - 7} \).

2. **Vertically stretching the result by a factor of 3**:
   - To vertically stretch a function by a factor of \( k \), you multiply the function by \( k \).
   - So, the function after the vertical stretch is \( g(x) = 3 \cdot \sqrt{x - 7} \).

Thus, the formula for the function \( g(x) \) is:

\[
g(x) = 3\sqrt{x - 7}
\]

Would you like more details on these transformations or have any questions? Here are some related questions that could help deepen your understanding:

1. How would the formula change if the function were shifted to the left instead of the right?
2. What effect does a vertical compression (rather than a stretch) have on the function?
3. How would the formula be affected if the function were reflected over the x-axis?
4. Can you describe the domain of the function \( g(x) = 3\sqrt{x - 7} \)?
5. How does the transformation affect the graph of \( f(x) = \sqrt{x} \) visually?

**Tip:** When dealing with function transformations, always apply horizontal shifts before vertical transformations for accurate results.

Explore how to transform the square root function \( f(x) = \sqrt{x} \) by shifting it 7 units to the right and vertically stretching the result by a factor of 3. Understand the step-by-step process and formula derivation for function \( g(x) \). Suitable for high school students studying function transformations.

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