If `f(x) = x^3 + 3x^2 + 12x - 2 sin x,` where `f: R rarr R,` then (A) f(x) is many-one and onto (B) f(x) is one-one and onto(C) f(x) is one-one and into (D) f(x) is many-one and into

Options

A. f(x) is many-one and onto
B. f(x) is one-one and onto
C. f(x) is one-one and into
D. f(x) is many-one and into

Answer

To determine the nature of the function \( f(x) = x^3 + 3x^2 + 12x - 2 \sin x \), we will analyze its properties step by step.

### Step 1: Identify the function type
The function \( f(x) \) is a combination of a polynomial and a trigonometric function. The polynomial part is \( x^3 + 3x^2 + 12x \), which is a cubic polynomial, and the trigonometric part is \( -2 \sin x \).

**Hint:** Recognize that cubic polynomials have specific behaviors regarding their limits and continuity.

### Step 2: Determine the limits as \( x \) approaches infinity and negative infinity
As \( x \to \infty \):
- The dominant term \( x^3 \) will cause \( f(x) \to \infty \).

As \( x \to -\infty \):
- Again, the dominant term \( x^3 \) will cause \( f(x) \to -\infty \).

**Hint:** Analyze the leading term of the polynomial to understand the behavior of the function at the extremes.

### Step 3: Check the range of the function
Since \( f(x) \) tends to \( \infty \) as \( x \to \infty \) and to \( -\infty \) as \( x \to -\infty \), it suggests that the function can take all real values. Therefore, the range of \( f(x) \) is \( \mathbb{R} \).

**Hint:** The behavior of the function at the extremes often indicates whether it is onto.

### Step 4: Differentiate the function to check for monotonicity
To check if \( f(x) \) is one-one, we differentiate it:
\[ f'(x) = 3x^2 + 6x + 12 - 2 \cos x \]

**Hint:** Finding the derivative helps us understand whether the function is increasing or decreasing.

### Step 5: Analyze the derivative
The quadratic part \( 3x^2 + 6x + 12 \) is always positive because its discriminant \( (6^2 - 4 \cdot 3 \cdot 12) = 36 - 144 = -108 \) is negative, meaning it has no real roots and opens upwards. The term \( -2 \cos x \) oscillates between -2 and 2.

Thus, \( f'(x) \) is always positive:
\[ f'(x) \geq 3x^2 + 6x + 10 > 0 \quad \text{for all } x \in \mathbb{R} \]

**Hint:** A positive derivative indicates that the function is strictly increasing.

### Step 6: Conclusion about the function
Since \( f(x) \) is strictly increasing and its range is \( \mathbb{R} \), we conclude that \( f(x) \) is one-one (injective) and onto (surjective).

**Final Answer:** The correct option is (B) \( f(x) \) is one-one and onto.