Understanding the Greatest Integer Function f(x) = [x]

The greatest integer function, denoted by f(x) = [x], maps a real number x to the greatest integer less than or equal to x. For example:

• [3.7] = 3
• [5] = 5
• [-2.3] = -3
• [0.9] = 0

The domain of this function is R (all real numbers) and the codomain is also given as R (all real numbers).

Analyzing if f(x) = [x] is One-One

A function f: A → B is said to be One-One (or injective) if distinct elements in the domain A have distinct images in the codomain B. In other words, if $f(x_1) = f(x_2)$, then it must imply $x_1 = x_2$ for all $x_1, x_2$ in the domain.

Let's consider the greatest integer function f(x) = [x].

• Take $x_1 = 1.5$. Then $f(x_1) = [1.5] = 1$.
• Take $x_2 = 1.9$. Then $f(x_2) = [1.9] = 1$.

Here, we have $f(1.5) = f(1.9) = 1$, but $1.5 \neq 1.9$. Since two different elements in the domain (1.5 and 1.9) have the same image (1) in the codomain, the function f(x) = [x] is not One-One.

Analyzing if f(x) = [x] is Onto

A function f: A → B is said to be 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. This means the range of the function must be equal to its codomain.

The range of the greatest integer function f(x) = [x] consists of all the possible values that f(x) can take. As we saw from the definition, the output of the greatest integer function is always an integer.

• For any integer k, if we take x such that $k \le x < k+1$, then $[x] = k$. This shows that every integer is in the range.

So, the range of f(x) = [x] is the set of all integers, denoted by Z.

The codomain of the function is given as R (all real numbers).

Since the range (Z) is a subset of the codomain (R), and Z is not equal to R (for example, real numbers like 0.5, π, $\sqrt{2}$ are in R but not in Z, so they are not in the range of f(x) = [x]), there are elements in the codomain that do not have a pre-image in the domain. Therefore, the function f(x) = [x] is not Onto.

Conclusion about f(x) = [x]

Based on the analysis:

• The function f(x) = [x] is not One-One because different real numbers can map to the same integer value.
• The function f(x) = [x] is not Onto because its range is the set of integers (Z), which is a proper subset of the codomain (R).

Therefore, the greatest integer function f : R → R given by f(x) = [x] is neither One-One nor onto.