Question

Let \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3} \), then \( f(x) \) is

• A. one-one and onto
• B. one-one and into
• C. many-one and into
• D. many-one and onto

Hint:

Any continuous function \(f: \mathbb{R} \to \mathbb{R}\) that has the same horizontal asymptote at both \(\pm \infty\) is guaranteed to be many-one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To check if a function is many-one, we check its derivative or see if it attains the same value at different points. To check if it is onto, we compare the range of the function with its codomain (\(\mathbb{R}\)).

Step 2: Key Formula or Approach:
For a rational function \(f(x) = \frac{ax^2+bx+c}{dx^2+ex+f}\), the range is typically a bounded interval if the denominator is never zero. If the range is not \(\mathbb{R}\), the function is "into."

Step 3: Detailed Explanation:
1. Denominator check: For \(3x^2 + x + 3\), the discriminant \(D = 1^2 - 4(3)(3) = -35<0\). The denominator is always positive, so the function is continuous. 2. Horizontal Asymptote: As \(x \to \infty\), \(f(x) \to 2/3\). This suggests the function is bounded. 3. Many-one: Since the function is continuous and returns to the same value \(2/3\) at both \(\infty\) and \(-\infty\), by Rolle's Theorem, it must have a turning point and repeat values. Thus, it is many-one. 4. Into: A quadratic-over-quadratic function has a range \([y_{min}, y_{max}]\). Since this range is a subset of \(\mathbb{R}\) and does not cover all real numbers, it is into.

Step 4: Final Answer:
The function is many-one and into.