Question

Let [x] represents the greatest integer which is less than or equal to x. Then the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = [x]\) will be

• A. One-one and onto
• B. One-one, but not onto
• C. Onto, but not one-one
• D. Neither one-one nor onto

Hint:

The graph of the greatest integer function is a step function. You can use the Horizontal Line Test to quickly check for injectivity and surjectivity. A horizontal line intersects the graph at multiple points (in fact, infinitely many points for integer y-values), so it's not one-one. The graph only covers integer y-values, not the entire real plane, so it's not onto.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to analyze the properties of the greatest integer function, also known as the floor function.
A function is one-one (injective) if different inputs produce different outputs. That is, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\).
A function is onto (surjective) if its range is equal to its codomain. For \(f: \mathbb{R} \to \mathbb{R}\), the range must be the entire set of real numbers \(\mathbb{R}\).
Step 2: Detailed Explanation:
Checking for One-one (Injectivity):
Let's take two different inputs, say \(x_1 = 2.3\) and \(x_2 = 2.8\).
For \(x_1 = 2.3\), the output is \(f(2.3) = [2.3] = 2\).
For \(x_2 = 2.8\), the output is \(f(2.8) = [2.8] = 2\).
Here, we have \(x_1 \neq x_2\) but \(f(x_1) = f(x_2)\).
Since different inputs map to the same output, the function is not one-one.
Checking for Onto (Surjectivity):
The function is defined as \(f: \mathbb{R} \to \mathbb{R}\), so its codomain is the set of all real numbers (\(\mathbb{R}\)).
The output of the greatest integer function \(f(x) = [x]\) is always an integer.
The range of the function is the set of all integers, \(\mathbb{Z}\).
Since the range (\(\mathbb{Z}\)) is not equal to the codomain (\(\mathbb{R}\)), the function is not onto. For example, there is no real number \(x\) such that \(f(x) = [x] = 2.5\).
Step 3: Final Answer:
The function \(f(x) = [x]\) is neither one-one nor onto.