If `A={x:f (x) =0} and B ={x: g(x) =0}," then "A cap B` will be the set of roots of the equation

Options

A. `(f(x))/(g(x))=0`
B. `(g(x))/(f(x))=0`
C. `[f(x)]^(2)+[g(x)]^(2)=0`
D. none

Answer

To solve the problem, we need to understand the sets A and B defined as follows:

- Set A contains the elements \( x \) such that \( f(x) = 0 \).
- Set B contains the elements \( x \) such that \( g(x) = 0 \).

The intersection of these two sets, denoted as \( A \cap B \), will contain the elements that are common to both sets, meaning the elements that satisfy both \( f(x) = 0 \) and \( g(x) = 0 \).

### Step-by-Step Solution:

1. **Define the Sets**:
- Let \( A = \{ x : f(x) = 0 \} \)
- Let \( B = \{ x : g(x) = 0 \} \)

2. **Understand Intersection**:
- The intersection \( A \cap B \) consists of all elements \( x \) such that both \( f(x) = 0 \) and \( g(x) = 0 \) hold true.
- Mathematically, \( A \cap B = \{ x : f(x) = 0 \text{ and } g(x) = 0 \} \)

3. **Identify Roots**:
- The elements in \( A \cap B \) are the common roots of the equations \( f(x) = 0 \) and \( g(x) = 0 \).
- Therefore, \( A \cap B \) will be the set of roots of the equation formed by combining both functions.

4. **Conclusion**:
- The set \( A \cap B \) will be the set of roots of the equation \( f(x) = 0 \) and \( g(x) = 0 \).

### Final Answer:
Thus, \( A \cap B \) will be the set of roots of the equations \( f(x) = 0 \) and \( g(x) = 0 \).