Question

The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \[ f(x) = (x - 1)(x - 2)(x - 3) \] is:

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

Hint:

A cubic function like \( f(x) = (x - 1)(x - 2)(x - 3) \) is onto but not one-one due to the turning points of the graph.

Answer 1

The Correct Option is B

Step 1: Understanding the function.
The given function is a cubic polynomial: \[ f(x) = (x - 1)(x - 2)(x - 3), \] which is a degree-3 polynomial. To determine if this function is one-one or onto, we need to analyze its properties.
Step 2: Checking if the function is one-one.
A function is said to be one-one (injective) if every element in the domain maps to a unique element in the codomain. In the case of a cubic polynomial like \( f(x) \), since the polynomial has a degree of 3, it is not one-one because it will have repeated values for different \( x \)-values due to the turning points in the graph. This can be confirmed by analyzing the derivative, which will show that the function has local maxima and minima.
Step 3: Checking if the function is onto.
A function is onto (surjective) if every element in the codomain has a pre-image in the domain. Since this is a cubic polynomial, as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \) also tends to infinity or negative infinity. Hence, for every real value of \( y \), there exists a real value of \( x \) such that \( f(x) = y \), making the function onto.
Step 4: Conclusion.
Since the function is onto but not one-one, the correct answer is (b).