The set of parameter 'a' for which the functions `f:R to R"defined by"f(x)=ax+sin x` is bijective, is

Options

A. [-1,1]
B. R-[-1,1]
C. R-[-1,1]
D. [-1,1]

Explanation

To determine the set of parameters \( a \) for which the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = ax + \sin x \) is bijective, we need to ensure that the function is either strictly increasing or strictly decreasing.

### Step-by-Step Solution:

1. **Understanding Bijective Functions**:
- A function is bijective if it is both one-to-one (injective) and onto (surjective). For a function to be one-to-one, it must be either strictly increasing or strictly decreasing.

2. **Finding the Derivative**:
- We start by calculating the derivative of the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(ax + \sin x) = a + \cos x \]

3. **Analyzing the Derivative**:
- For \( f(x) \) to be strictly increasing, we need:
\[ f'(x) > 0 \quad \text{for all } x \]
- For \( f(x) \) to be strictly decreasing, we need:
\[ f'(x) < 0 \quad \text{for all } x \]

4. **Finding Conditions for Strictly Increasing**:
- The cosine function \( \cos x \) oscillates between -1 and 1. Therefore, the minimum value of \( \cos x \) is -1.
- Thus, for \( f'(x) \) to be greater than 0:
\[ a + \cos x > 0 \quad \Rightarrow \quad a + (-1) > 0 \quad \Rightarrow \quad a > 1 \]

5. **Finding Conditions for Strictly Decreasing**:
- Similarly, for \( f'(x) \) to be less than 0:
\[ a + \cos x < 0 \quad \Rightarrow \quad a + 1 < 0 \quad \Rightarrow \quad a < -1 \]

6. **Combining Conditions**:
- From the analysis, we find two intervals:
- \( a > 1 \) (strictly increasing)
- \( a < -1 \) (strictly decreasing)

7. **Conclusion**:
- Therefore, the set of parameters \( a \) for which the function \( f(x) = ax + \sin x \) is bijective is:
\[ a \in (-\infty, -1) \cup (1, \infty) \]
- This means \( a \) can take any real value except those in the interval \([-1, 1]\).

### Final Answer:
The set of values for the parameter \( a \) is:
\[ \mathbb{R} \setminus [-1, 1] \]