If the function f(x)=[3.5+bsin x] (where [.] denotes the greatest integer function) is an even function, the complete set of values of b is

Options

A. (-0.5,0.5)
B. [-0.5,0.5]
C. (0,1)
D. [-1,1]

Answer

To solve the problem, we need to determine the complete set of values of \( b \) such that the function \( f(x) = [3.5 + b \sin x] \) is an even function.

### Step-by-Step Solution:

1. **Understanding Even Functions**:
An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \).

2. **Setting Up the Equation**:
For our function, we have:
\[ f(-x) = [3.5 + b \sin(-x)] = [3.5 - b \sin x] \]
Since \( \sin(-x) = -\sin x \).

3. **Equating the Two Expressions**:
For \( f(x) \) to be even, we need:
\[ [3.5 + b \sin x] = [3.5 - b \sin x] \]

4. **Analyzing the Greatest Integer Function**:
The greatest integer function \( [x] \) returns the largest integer less than or equal to \( x \). Therefore, we need to consider the values of \( 3.5 + b \sin x \) and \( 3.5 - b \sin x \).

5. **Finding the Range of the Function**:
- The term \( b \sin x \) varies between \( -|b| \) and \( |b| \) as \( \sin x \) varies from -1 to 1.
- Thus, \( 3.5 + b \sin x \) will range from \( 3.5 - |b| \) to \( 3.5 + |b| \).

6. **Setting the Conditions for Equality**:
For the two expressions to be equal under the greatest integer function, the ranges must overlap in such a way that they yield the same integer values. This means:
\[ 3.5 - |b| < 4 \quad \text{and} \quad 3.5 + |b| > 3 \]

7. **Solving the Inequalities**:
- From \( 3.5 - |b| < 4 \):
\[ -|b| < 0.5 \implies |b| < 0.5 \]
- From \( 3.5 + |b| > 3 \):
\[ |b| > -0.5 \quad \text{(which is always true since } |b| \geq 0 \text{)} \]

8. **Combining the Results**:
Therefore, the only condition we have is \( |b| < 0.5 \), which can be expressed as:
\[ -0.5 < b < 0.5 \]

9. **Final Answer**:
The complete set of values of \( b \) is:
\[ b \in (-0.5, 0.5) \]