The function of `f:R to R`, defined by `f(x)=[x]`, where [x] denotes the greatest integer less than or equal to x, is

Options

A. one-one
B. onto
C. one-one and onto
D. neither one-one nor onto

Answer

To analyze the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = [x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will determine whether this function is one-one (injective) and onto (surjective).

### Step 1: Understanding the Function
The function \( f(x) = [x] \) gives the greatest integer less than or equal to \( x \). For example:
- If \( x = 2.3 \), then \( f(2.3) = 2 \).
- If \( x = -1.7 \), then \( f(-1.7) = -2 \).
- If \( x = 3 \), then \( f(3) = 3 \).

### Step 2: Determine if the Function is One-One
A function is one-one if different inputs produce different outputs. Let's check this property:
- For \( x_1 = 0.5 \) and \( x_2 = 0.9 \), we have:
\[ f(0.5) = [0.5] = 0 \quad \text{and} \quad f(0.9) = [0.9] = 0 \]
Here, \( f(0.5) = f(0.9) \) but \( 0.5 \neq 0.9 \).

Since multiple inputs can yield the same output, the function is **not one-one**.

### Step 3: Determine if the Function is Onto
A function is onto if every element in the codomain has a pre-image in the domain. The codomain here is \( \mathbb{R} \), while the range of \( f(x) \) is the set of integers \( \mathbb{Z} \) (since \( f(x) \) can only take integer values).

- For example, there is no \( x \in \mathbb{R} \) such that \( f(x) = 0.5 \) or any other non-integer value.

Since not every real number is an output of the function, \( f \) is **not onto**.

### Conclusion
The function \( f(x) = [x] \) is neither one-one nor onto.

### Final Answer
The function \( f(x) = [x] \) is neither one-one nor onto.

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