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

Options

A. `{f(x)}^(2)+{g(x)}^(2) = 0`
B. `(f(x))/(g(x))`
C. `(g(x))/(f(x))`
D. none of these

Answer

To solve the problem, we need to determine the set of roots for the union of two sets A and B, where:

- \( A = \{ x : f(x) = 0 \} \)
- \( B = \{ x : g(x) = 0 \} \)

The union of these two sets, denoted as \( A \cup B \), will contain all the roots of the equations defined by \( f(x) \) and \( g(x) \).

### Step-by-Step Solution:

1. **Define the Functions**:
Let's assume:
\[ f(x) = (x - A)(x - B) \]
\[ g(x) = (x - B)(x - C) \]
Here, \( A \), \( B \), and \( C \) are constants representing the roots of the respective functions.

2. **Find the Roots of Each Function**:
- The roots of \( f(x) = 0 \) are \( x = A \) and \( x = B \). Thus, the set \( A \) is:
\[ A = \{ A, B \} \]
- The roots of \( g(x) = 0 \) are \( x = B \) and \( x = C \). Thus, the set \( B \) is:
\[ B = \{ B, C \} \]

3. **Union of the Sets**:
Now, we find the union of sets \( A \) and \( B \):
\[ A \cup B = \{ A, B \} \cup \{ B, C \} \]
This results in:
\[ A \cup B = \{ A, B, C \} \]

4. **Conclusion**:
The union \( A \cup B \) represents the set of roots of the equation formed by the product of the two functions:
\[ f(x) \cdot g(x) = 0 \]
Therefore, the roots of the combined equation will be \( A \), \( B \), and \( C \).

### Final Answer:
The set \( A \cup B \) will be the set of roots of the equation \( f(x) \cdot g(x) = 0 \).