In the xy-plane, the graph of the functions g has zeros at -4, 2, and 4. Which of the following could define g?

Options

A. `g(x)=(x-4)(x-2)(x+4)`
B. `g(x)=(x-4)^(2)(x-2)`
C. `g(x)=(x-4)(x+2)(x+4)`
D. `g(x)=(x+2)(x+4)^(2)`

Answer

To solve the problem, we need to find a function \( g(x) \) that has zeros at \( x = -4 \), \( x = 2 \), and \( x = 4 \). This means that when we substitute these values into the function \( g(x) \), the output should be zero.

### Step-by-step Solution:

1. **Understanding Zeros of a Function**:
- A function \( g(x) \) has a zero at \( x = a \) if \( g(a) = 0 \). Given that \( g \) has zeros at \( -4 \), \( 2 \), and \( 4 \), we can express this mathematically.

2. **Formulating the Function**:
- The zeros of the function indicate that \( g(x) \) can be expressed as a product of factors corresponding to these zeros. Specifically, if \( g \) has zeros at \( x = -4 \), \( x = 2 \), and \( x = 4 \), we can write:
\[ g(x) = k(x + 4)(x - 2)(x - 4) \]
- Here, \( k \) is a constant that can be any non-zero value (often set to 1 for simplicity).

3. **Writing the Function**:
- For simplicity, let’s assume \( k = 1 \). Thus, we can write:
\[ g(x) = (x + 4)(x - 2)(x - 4) \]

4. **Expanding the Function**:
- We can expand this function if needed, but for the purpose of identifying the function that defines \( g \), it is sufficient to leave it in factored form.

5. **Conclusion**:
- Therefore, one possible definition of \( g \) that has zeros at the specified points is:
\[ g(x) = (x + 4)(x - 2)(x - 4) \]