The set of value of a for which the function `f(x)=sinx+[(x^(2))/(a)]` defined on [-2,2] lies an odd function , is

Options

A. `(4,oo)`
B. `[-4,4]`
C. `(-oo,4)`
D. none of these

Answer

To determine the set of values of \( a \) for which the function \( f(x) = \sin x + \left\lfloor \frac{x^2}{a} \right\rfloor \) defined on the interval \([-2, 2]\) is an odd function, we will follow these steps:

### Step 1: Definition of Odd Function
A function \( f(x) \) is considered odd if it satisfies the condition:
\[ f(-x) = -f(x) \]
for all \( x \) in its domain.

### Step 2: Calculate \( f(-x) \)
We start by calculating \( f(-x) \):
\[ f(-x) = \sin(-x) + \left\lfloor \frac{(-x)^2}{a} \right\rfloor \]
Using the property of sine, \( \sin(-x) = -\sin(x) \), we have:
\[ f(-x) = -\sin(x) + \left\lfloor \frac{x^2}{a} \right\rfloor \]

### Step 3: Set Up the Odd Function Condition
Now, we can set up the equation based on the odd function condition:
\[ f(-x) = -f(x) \]
Substituting the expressions we have:
\[ -\sin(x) + \left\lfloor \frac{x^2}{a} \right\rfloor = -\left( \sin(x) + \left\lfloor \frac{x^2}{a} \right\rfloor \right) \]
This simplifies to:
\[ -\sin(x) + \left\lfloor \frac{x^2}{a} \right\rfloor = -\sin(x) - \left\lfloor \frac{x^2}{a} \right\rfloor \]

### Step 4: Simplifying the Equation
Adding \( \sin(x) \) to both sides gives:
\[ \left\lfloor \frac{x^2}{a} \right\rfloor + \left\lfloor \frac{x^2}{a} \right\rfloor = 0 \]
This implies:
\[ 2\left\lfloor \frac{x^2}{a} \right\rfloor = 0 \]
Thus, we have:
\[ \left\lfloor \frac{x^2}{a} \right\rfloor = 0 \]

### Step 5: Conditions for the Floor Function
The condition \( \left\lfloor \frac{x^2}{a} \right\rfloor = 0 \) means:
\[ 0 \leq \frac{x^2}{a} < 1 \]
This can be rewritten as:
\[ 0 \leq x^2 < a \]

### Step 6: Determine the Range of \( x^2 \)
Since \( x \) is in the interval \([-2, 2]\), the maximum value of \( x^2 \) is:
\[ x^2 \leq 4 \]
Thus, we require:
\[ x^2 < a \implies a > 4 \]

### Conclusion
The set of values of \( a \) for which the function \( f(x) \) is an odd function is:
\[ a > 4 \]