Greatest Integer Function (GIF)

Greatest Integer Function (GIF)

The Greatest Integer Function, also known as the floor function, is denoted by $\lfloor x \rfloor$ and is defined as the greatest integer less than or equal to $x$. In other words, it rounds $x$ down to the nearest integer.

Definition

For any real number $x$, the Greatest Integer Function $\lfloor x \rfloor$ is defined as the largest integer that is less than or equal to $x$. Mathematically, it can be expressed as:

$$ \lfloor x \rfloor = n, \quad \text{where } n \leq x < n+1 \text{ and } n \text{ is an integer} $$

Properties of GIF

The following table summarizes the key properties of the Greatest Integer Function:

Property Description
Discontinuity GIF is discontinuous at all integer points.
Range The range of GIF is the set of all integers, $\mathbb{Z}$.
Periodicity GIF is not periodic.
Monotonicity GIF is non-decreasing; it remains constant or increases as $x$ increases.

Graph of GIF

The graph of the Greatest Integer Function is a series of steps. It remains constant at the value of $\lfloor x \rfloor$ until the next integer is reached, at which point it jumps up to the next integer.

Examples

Let's illustrate the Greatest Integer Function with some examples:

  1. $\lfloor 3.5 \rfloor = 3$
  2. $\lfloor -1.8 \rfloor = -2$
  3. $\lfloor 7 \rfloor = 7$
  4. $\lfloor -2 \rfloor = -2$

Formulas Involving GIF

Here are some useful formulas and identities involving the Greatest Integer Function:

  1. $\lfloor x \rfloor + \lfloor -x \rfloor = \begin{cases} -1, & \text{if } x \text{ is not an integer} \ x, & \text{if } x \text{ is an integer} \end{cases}$

  2. For any integer $n$ and real number $x$, $\lfloor x+n \rfloor = \lfloor x \rfloor + n$

  3. For any real numbers $x$ and $y$, $\lfloor x \rfloor + \lfloor y \rfloor \leq \lfloor x+y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor + 1$

Applications in Exams

In exams, you may encounter problems that require you to evaluate expressions involving the Greatest Integer Function, solve inequalities, or analyze the behavior of a function that includes GIF.

Example Problem

Evaluate the expression $\lfloor 2.7 \rfloor + \lfloor -3.4 \rfloor$.

Solution:

$\lfloor 2.7 \rfloor = 2$ (since 2 is the greatest integer less than or equal to 2.7)

$\lfloor -3.4 \rfloor = -4$ (since -4 is the greatest integer less than or equal to -3.4)

Therefore, $\lfloor 2.7 \rfloor + \lfloor -3.4 \rfloor = 2 + (-4) = -2$.

Practice Problems

  1. Find the value of $\lfloor 5.9 \rfloor - \lfloor -2.1 \rfloor$.
  2. Solve the inequality $\lfloor x \rfloor > 3$.
  3. If $f(x) = \lfloor x \rfloor + \lfloor x+0.5 \rfloor$, find the value of $f(3.5)$.

Understanding the Greatest Integer Function is essential for solving a variety of problems in mathematics, especially in calculus, number theory, and discrete mathematics. Remember that practice is key to mastering the application of GIF in different contexts.