Question

The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 + x \) is:

• A. One-one
• B. Onto
• C. Many-one
• D. None of these

Hint:

Quadratic functions are typically many-to-one because they are not injective (one-to-one), meaning multiple inputs can yield the same output.

Answer 1

The Correct Option is C

Step 1: The function \( f(x) = x^2 + x \) is a quadratic function. A quadratic function is not one-to-one because different values of \( x \) can yield the same output. For example, \( f(-1) = f(0) = 0 \), showing that the function is many-to-one.
Step 2: The function is not onto because not every real number can be obtained as the value of \( f(x) \). The range of \( f(x) = x^2 + x \) is \( \left[ -\frac{1}{4}, \infty \right) \), and negative values less than \( -\frac{1}{4} \) cannot be obtained.