If f and g are two functions defined as f (x) = x + 2, `x le ` 0 , g (x) = 3 , x `ge` 0, then the domain of f + g is

Options

A. {0}
B. `[0, infty)`
C. `(- infty, infty)`
D. `( - infty, 0 )`

Answer

To find the domain of the function \( f + g \), we need to analyze the domains of the individual functions \( f(x) \) and \( g(x) \).

1. **Identify the domain of \( f(x) \)**:
- The function \( f(x) = x + 2 \) is defined for \( x \leq 0 \).
- Therefore, the domain of \( f \) is \( (-\infty, 0] \).

2. **Identify the domain of \( g(x) \)**:
- The function \( g(x) = 3 \) is defined for \( x \geq 0 \).
- Therefore, the domain of \( g \) is \( [0, \infty) \).

3. **Find the intersection of the domains**:
- The domain of \( f + g \) is the intersection of the domains of \( f \) and \( g \).
- The domain of \( f \) is \( (-\infty, 0] \) and the domain of \( g \) is \( [0, \infty) \).
- The intersection of these two domains is the set of values that are common to both, which is \( \{0\} \).

4. **Conclusion**:
- Therefore, the domain of \( f + g \) is \( \{0\} \) or in interval notation, it can be written as \( [0, 0] \).

### Final Answer:
The domain of \( f + g \) is \( \{0\} \).

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