Let `f:A->B` be a function defined by `f(x) =sqrt3sin x +cos x+4.` If `f` is invertible, then

Options

A. `A=[-2pi//3, pi//3], B=[2,6]`
B. `A=[pi//6, 5pi//6], B=[-2,2]`
C. `A=[-pi//2, pi//2], B=[2,6]`
D. `A=[-pi//3, pi//3], B=[2,6]`

Answer

To determine the values of \( a \) and \( b \) such that the function \( f(x) = \sqrt{3} \sin x + \cos x + 4 \) is invertible, we need to find the domain and range of the function.

### Step 1: Rewrite the Function
We start with the function:
\[ f(x) = \sqrt{3} \sin x + \cos x + 4 \]

### Step 2: Simplify the Trigonometric Expression
We can express \( \sqrt{3} \sin x + \cos x \) in a single sine function. We can use the sine addition formula:
\[ R \sin(x + \phi) = R \sin x \cos \phi + R \cos x \sin \phi \]
where \( R = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 \) and \( \phi = \tan^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6} \).

Thus, we can rewrite:
\[ \sqrt{3} \sin x + \cos x = 2 \sin\left(x + \frac{\pi}{6}\right) \]
So, we have:
\[ f(x) = 2 \sin\left(x + \frac{\pi}{6}\right) + 4 \]

### Step 3: Determine the Range of the Function
The sine function \( \sin(x + \frac{\pi}{6}) \) oscillates between -1 and 1. Therefore:
\[ 2 \sin\left(x + \frac{\pi}{6}\right) \text{ ranges from } -2 \text{ to } 2 \]
Adding 4 shifts this range:
\[ f(x) \text{ ranges from } -2 + 4 = 2 \text{ to } 2 + 4 = 6 \]
Thus, the range of \( f(x) \) is:
\[ [2, 6] \]

### Step 4: Determine the Co-domain
Since the function \( f \) is invertible, it must be both one-to-one and onto. This means the range of \( f \) must equal the co-domain \( B \). Therefore, we have:
\[ b = [2, 6] \]

### Step 5: Determine the Domain of the Function
For \( f \) to be invertible, we need to find the domain \( A \). The range of \( f^{-1} \) will be the same as the domain of \( f \). Since \( f \) is defined for all real numbers \( x \), the domain \( A \) can be any interval that allows the function to remain one-to-one.

To ensure \( f \) is one-to-one, we can restrict the domain to an interval where the sine function does not repeat values. A suitable choice is:
\[ a = \left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right] \]

### Final Answer
Thus, we conclude:
\[ a = \left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right], \quad b = [2, 6] \]