The function `f:RrarrR` defined by `f(x)=(x-1)(x-2)(x-3)` is :

Options

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

Answer

To determine the nature of the function \( f(x) = (x-1)(x-2)(x-3) \), we need to analyze whether it is one-to-one (1-1) and/or onto (onto).

### Step 1: Check if the function is one-to-one (1-1)
A function is one-to-one if different inputs produce different outputs. In other words, if \( f(a) = f(b) \) implies that \( a = b \).

1. **Evaluate the function at specific points:**
- Calculate \( f(1) \):
\[ f(1) = (1-1)(1-2)(1-3) = 0 \cdot (-1) \cdot (-2) = 0 \]
- Calculate \( f(2) \):
\[ f(2) = (2-1)(2-2)(2-3) = 1 \cdot 0 \cdot (-1) = 0 \]
- Calculate \( f(3) \):
\[ f(3) = (3-1)(3-2)(3-3) = 2 \cdot 1 \cdot 0 = 0 \]

2. **Conclusion from evaluations:**
- We find that \( f(1) = f(2) = f(3) = 0 \). Since three different inputs (1, 2, and 3) give the same output (0), the function is not one-to-one.

### Step 2: Check if the function is onto (onto)
A function is onto if every possible output in the codomain (in this case, all real numbers) is achieved by some input from the domain.

1. **Analyze the nature of the polynomial:**
- The function \( f(x) = (x-1)(x-2)(x-3) \) is a cubic polynomial. The general behavior of cubic functions is that they can take on all real values as \( x \) approaches positive or negative infinity.

2. **Conclusion for onto:**
- Since \( f(x) \) is a cubic polynomial, it will cover all real numbers as outputs. Therefore, the function is onto.

### Final Conclusion
- The function \( f(x) = (x-1)(x-2)(x-3) \) is not one-to-one but is onto.

### Answer
The correct option is: **Option 2: onto but not one-to-one.**

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