Understanding the Greatest Integer Function

The question asks us to analyze the properties of the greatest integer function, denoted by \( f(x) = \lfloor x \rfloor \). This function takes any real number \( x \) and returns the largest integer that is less than or equal to \( x \).

Let's examine each statement given in the options to determine which ones are correct for the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \lfloor x \rfloor \).

Analyzing Statement (a): \( f \) is one-one

A function \( f: A \to B \) is one-one (or injective) if for any two distinct elements \( x_1, x_2 \) in the domain \( A \), their images under \( f \) are also distinct, i.e., if \( x_1 \neq x_2 \), then \( f(x_1) \neq f(x_2) \). Equivalently, if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

Let's consider the greatest integer function \( f(x) = \lfloor x \rfloor \). Take two different real numbers, for example, \( x_1 = 2.1 \) and \( x_2 = 2.5 \). These are distinct numbers, \( 2.1 \neq 2.5 \).

• \( f(2.1) = \lfloor 2.1 \rfloor = 2 \)
• \( f(2.5) = \lfloor 2.5 \rfloor = 2 \)

Here, \( f(2.1) = f(2.5) = 2 \), even though \( 2.1 \neq 2.5 \). Since different inputs can give the same output, the function \( f \) is not one-one.

Therefore, statement (a) is incorrect.

Analyzing Statement (b): \( f \) is not onto

A function \( f: A \to B \) is onto (or surjective) if for every element \( y \) in the codomain \( B \), there exists at least one element \( x \) in the domain \( A \) such that \( f(x) = y \).

The domain of \( f \) is \( \mathbb{R} \) (all real numbers), and the codomain is also \( \mathbb{R} \) (all real numbers). The output of the greatest integer function \( \lfloor x \rfloor \) is always an integer.

Consider an element \( y \) in the codomain \( \mathbb{R} \) that is not an integer, for example, \( y = 0.5 \). Can we find a real number \( x \) such that \( f(x) = \lfloor x \rfloor = 0.5 \)? No, because the output of the greatest integer function is always an integer. There is no real number \( x \) whose greatest integer less than or equal to it is 0.5.

Since there are elements in the codomain \( \mathbb{R} \) (like 0.5, 1.7, \(\pi\)) that are not in the range of \( f \), the function \( f \) is not onto.

Therefore, statement (b) is correct.

Analyzing Statement (c): Range of \( f \) is \( \mathbb{I} \) (set of integers)

The range of a function is the set of all possible output values. For \( f(x) = \lfloor x \rfloor \), the definition states that the output is the greatest integer less than or equal to \( x \). This means the output value is always an integer.

Can any integer be an output? Yes. For any integer \( n \in \mathbb{Z} \), we can choose \( x = n \). Then \( f(n) = \lfloor n \rfloor = n \). So, every integer is a possible output of the function.

Thus, the set of all possible output values is the set of all integers, which is denoted by \( \mathbb{Z} \) or \( \mathbb{I} \).

Therefore, statement (c) is correct.

Analyzing Statement (d): \( f(2.5) = 2 \)

We need to evaluate the function \( f(x) = \lfloor x \rfloor \) at \( x = 2.5 \). \( f(2.5) = \lfloor 2.5 \rfloor \).

The greatest integer less than or equal to 2.5 is 2.

\( \lfloor 2.5 \rfloor = 2 \).

Therefore, \( f(2.5) = 2 \), and statement (d) is correct.

Analyzing Statement (e): \( f \) is bijective

A function is bijective if it is both one-one and onto. From our analysis above, we found that the greatest integer function \( f(x) = \lfloor x \rfloor \) is neither one-one (statement a is incorrect) nor onto (statement b is correct, meaning it is not onto).

Since \( f \) is not one-one and not onto, it cannot be bijective.

Therefore, statement (e) is incorrect.

Summary of Statement Analysis

Based on our analysis, statements (b), (c), and (d) are correct.

Conclusion

The correct statements about the greatest integer function \( f(x) = \lfloor x \rfloor \) from \( \mathbb{R} \) to \( \mathbb{R} \) are (b) \( f \) is not onto, (c) Range of \( f \) is \( \mathbb{I} \), and (d) \( f(2.5) = 2 \).

This corresponds to option 2, which lists B, C, D only.

Revision Table: Function Properties

Additional Information: Greatest Integer Function Details

The greatest integer function \( f(x) = \lfloor x \rfloor \) is also known as the floor function. It is a step function because its graph looks like a series of steps.

• For any integer \( n \), \( \lfloor n \rfloor = n \).
• For any non-integer \( x \), \( \lfloor x \rfloor < x \).
• The value of \( \lfloor x \rfloor \) is constant over any interval \( [n, n+1) \) for an integer \( n \). For instance, \( \lfloor x \rfloor = 3 \) for all \( x \) such that \( 3 \le x < 4 \).
• The function is discontinuous at every integer value.

Understanding the domain, codomain, and range is crucial when analyzing whether a function is one-one or onto. In this case, the codomain is given as \( \mathbb{R} \), while the range is the set of integers \( \mathbb{Z} \). Since the range is a proper subset of the codomain, the function is not onto.