The domain of f(x)=2 sinx + 3 cosx +4 is

Options

A. R
B. `R^+`
C. R-{0}
D. `R-{npi:n inI^+}`

Answer

To find the domain of the function \( f(x) = 2 \sin x + 3 \cos x + 4 \), we will follow these steps:

### Step 1: Identify the components of the function
The function consists of three parts: \( 2 \sin x \), \( 3 \cos x \), and the constant \( 4 \).

### Step 2: Determine the domain of each component
1. **For \( 2 \sin x \)**: The sine function, \( \sin x \), is defined for all real numbers. Therefore, the domain of \( 2 \sin x \) is \( x \in \mathbb{R} \).

2. **For \( 3 \cos x \)**: The cosine function, \( \cos x \), is also defined for all real numbers. Hence, the domain of \( 3 \cos x \) is \( x \in \mathbb{R} \).

3. **For the constant \( 4 \)**: A constant function is defined for all real numbers as well. Thus, the domain of \( 4 \) is \( x \in \mathbb{R} \).

### Step 3: Combine the domains
Since all three components \( 2 \sin x \), \( 3 \cos x \), and \( 4 \) have the same domain of \( x \in \mathbb{R} \), we take the intersection of these domains.

### Step 4: Conclusion
The intersection of the domains is still \( x \in \mathbb{R} \). Therefore, the domain of the function \( f(x) = 2 \sin x + 3 \cos x + 4 \) is:

\[ \text{Domain of } f(x) = \mathbb{R} \]