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We are given the function \( f \), where:

- \( f(3) = 0 \)
- \( f(6) = -4 \)

The new function \( g(x) \) is defined as \( g(x) = f(x - 4) \), and we need to find which value of \( x \) will make \( g(x) = 0 \).

### Step-by-Step Solution:

We know that \( g(x) = f(x - 4) \), and we are looking for when \( g(x) = 0 \). This means we need to solve the equation:

\[
g(x) = f(x - 4) = 0
\]

So, \( f(x - 4) = 0 \). From the given information, we know that \( f(3) = 0 \), so we need to make \( x - 4 = 3 \). Solving for \( x \):

\[
x - 4 = 3 \quad \Rightarrow \quad x = 7
\]

Thus, \( x = 7 \) must be a solution to \( g(x) = 0 \).

### Correct Answer:
The correct answer is **D** \( x = 7 \).

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Would you like further clarification or details on this?

Here are 5 related questions for further practice:

1. How would the solution change if \( f(4) = 0 \) instead of \( f(3) = 0 \)?
2. If \( g(x) = f(x - 5) \), which value of \( x \) would solve \( g(x) = 0 \) based on the same \( f \)?
3. What does shifting a function horizontally by \( k \) units mean in terms of its effect on the graph?
4. What happens if we change the function \( g(x) = f(2x - 8) \)? How would you find solutions?
5. How does the derivative of \( g(x) = f(x - 4) \) relate to the derivative of \( f(x) \)?

**Tip:** When solving for functions of the form \( g(x) = f(x - c) \), try to solve for the expression inside the function \( f \) to make it match known values.

For the function f, it is known that f(3) = 0 and f(6) = -4. The function g is given by g(x) = f(x - 4). Which of the following must be a solution to g(x) = 0?

Learn how to solve g(x) = f(x - 4) with the given conditions f(3) = 0 and f(6) = -4. This algebra problem demonstrates horizontal shifting of functions and helps students find solutions step-by-step. Suitable for Grades 8-10.

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