The function `f : R -> R` is 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 and onto
D. neither one-one nor onto

Answer

To analyze the function \( f(x) = (x-1)(x-2)(x-3) \) and determine if it is one-to-one or onto, we will follow these steps:

### Step 1: Determine if the function is one-to-one (injective)

A function is one-to-one if every element in the domain maps to a unique element in the codomain. This means that if \( f(a) = f(b) \), then \( a \) must equal \( b \).

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

Since \( f(1) = f(2) = f(3) = 0 \), we see that multiple inputs yield the same output (0). Therefore, the function is **not one-to-one**.

### Step 2: Determine if the function is onto (surjective)

A function is onto if every element in the codomain has a pre-image in the domain. In this case, we need to check if the range of \( f(x) \) is equal to the codomain, which is all real numbers \( \mathbb{R} \).

1. **Analyze the behavior of the function**:
- The function \( f(x) \) is a cubic polynomial, which means it can take on all real values as \( x \) approaches positive or negative infinity.
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
- As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( +\infty \).

2. **Check specific values**:
- For \( x = 0 \): \( f(0) = (0-1)(0-2)(0-3) = -6 \) (negative)
- For \( x = -2 \): \( f(-2) = (-2-1)(-2-2)(-2-3) = -15 \) (negative)
- For \( x = 4 \): \( f(4) = (4-1)(4-2)(4-3) = 6 \) (positive)

Since the function takes on both negative and positive values, and since it can achieve all values in between as well, we conclude that the range of \( f(x) \) is all real numbers.

### Conclusion

Since the range of \( f(x) \) is equal to the codomain \( \mathbb{R} \), the function is **onto**. However, since it is not one-to-one, we conclude:

- The function is **not one-to-one** but is **onto**.