The domain of `f(x)=2sinx+3cos x+4` is :

Options

A. R
B. `R^(+)`
C. `R-{0}`
D. `R-{npi:n in l^(+)}`

Answer

To find the domain of the function \( f(x) = 2\sin x + 3\cos x + 4 \), we need to determine the values of \( x \) for which the function is well-defined.

### Step-by-Step Solution:

1. **Understanding the Components of the Function**:
- The function \( f(x) \) is composed of the sine and cosine functions, which are periodic and defined for all real numbers.
- The sine function, \( \sin x \), and the cosine function, \( \cos x \), can take any real number as input.

2. **Analyzing the Sine and Cosine Functions**:
- Both \( \sin x \) and \( \cos x \) are defined for all \( x \in \mathbb{R} \).
- This means that there are no restrictions on the values of \( x \) that can be used in these functions.

3. **Combining the Functions**:
- The expression \( 2\sin x + 3\cos x + 4 \) is simply a linear combination of \( \sin x \) and \( \cos x \) with a constant term added.
- Since both \( \sin x \) and \( \cos x \) are defined for all real numbers, their linear combination will also be defined for all real numbers.

4. **Conclusion on the Domain**:
- Therefore, the domain of \( f(x) \) is all real numbers, which can be expressed as:
\[ \text{Domain of } f(x) = \mathbb{R} \]

### Final Answer:
The domain of \( f(x) = 2\sin x + 3\cos x + 4 \) is \( \mathbb{R} \) (all real numbers).

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