If `f: R to R ` be defined by `f(x) =(1)/(x), AA x in R.` Then , f is

Options

A. one-one
B. onto
C. bijective
D. f is not defined

Explanation

To determine the properties of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{1}{x} \), we will analyze its domain, whether it is one-to-one (injective), and whether it is onto (surjective).

### Step 1: Determine the Domain
The function \( f(x) = \frac{1}{x} \) is undefined when \( x = 0 \) because division by zero is not allowed. Therefore, the domain of the function is all real numbers except zero.

**Domain**: \( \mathbb{R} - \{0\} \)

### Step 2: Check if the Function is One-to-One (Injective)
A function is one-to-one if different inputs produce different outputs. To check this, we assume \( f(x_1) = f(x_2) \) for \( x_1, x_2 \in \mathbb{R} - \{0\} \).

\[ f(x_1) = f(x_2) \implies \frac{1}{x_1} = \frac{1}{x_2} \]

Cross-multiplying gives:

\[ x_2 = x_1 \]

Since \( x_1 \) must equal \( x_2 \), the function is one-to-one.

**Conclusion**: \( f \) is one-to-one.

### Step 3: Check if the Function is Onto (Surjective)
A function is onto if every element in the codomain has a pre-image in the domain. The codomain of our function is \( \mathbb{R} \).

To see if \( f \) is onto, we need to check if there exists an \( x \in \mathbb{R} - \{0\} \) for every \( y \in \mathbb{R} \) such that:

\[ f(x) = y \implies \frac{1}{x} = y \implies x = \frac{1}{y} \]

However, if \( y = 0 \), then \( x = \frac{1}{0} \) is undefined. Therefore, there is no \( x \) in the domain that maps to \( y = 0 \).

**Conclusion**: \( f \) is not onto.

### Final Conclusion
The function \( f(x) = \frac{1}{x} \) is one-to-one but not onto when considered from \( \mathbb{R} - \{0\} \) to \( \mathbb{R} \).

### Summary
- **Domain**: \( \mathbb{R} - \{0\} \)
- **One-to-One**: Yes
- **Onto**: No