Functions
1.0 Definitions
2.0 Relation
3.0 Types of Relation
4.0 Functions
5.0 Standard Real Functions and their Graphical Representation
5.10 Reciprocal Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
6.0 Operations on Real Functions
7.0 Types of Functions
8.0 Composition of a Function
9.0 Inverse of a Function
5.4 Greatest Integer Function or Floor Function
5.1 Constant Function
5.2 Identity Function
5.3 Modulus Function
5.4 Greatest Integer Function or Floor Function
5.5 Smallest Integer Function or Ceiling Function
5.6 Fractional Part Function
5.7 Signum Function
5.8 Exponential Function
5.9 Logarithmic Function
5.11 Square Root or Radical Function
5.12 Square Function
5.13 Cube Function
5.14 Cube Root Function
For any real number $x$, the greatest integer less than or equal to $x$ is denoted as, $$\left[ x \right]\;\;or\;\;\left\lfloor x \right\rfloor \;$$
Here, $$f(x) = \left[ x \right]\;,\;\forall \;x \in R$$
and is called the greatest integer function or floor function. This is a step function. For example:
| Value | Greatest Integer |
| $[1.9]$ | $1$ |
| $[-2.1]$ | $-3$ |
| $[2.1]$ | $2$ |
| $[2]$ | $2$ |
| $[1.1]$ | $1$ |
To plot the graph of the function, we can write the function as $$y = \left[ x \right]$$ Taking the intervals of $x$ and finding the value of $y$ as shown below.
| $y$ | $x$ |
| $-3$ | $ - 3 \leqslant x < - 2$ |
| $-2$ | $ - 2 \leqslant x < - 1$ |
| $-1$ | $ - 1 \leqslant x < 0$ |
| $0$ | $0 \leqslant x < 1$ |
| $1$ | $ 1 \leqslant x < 2$ |
| $2$ | $2 \leqslant x < 3$ |
| $3$ | $ 3 \leqslant x < 4$ |
Using the values we get in the table, plot the graph in the corresponding intervals of $x$.
Example 15. Find the value of $$\left[ {\frac{1}{2} + \frac{1}{{100}}} \right] + \left[ {\frac{1}{2} + \frac{2}{{100}}} \right] + \left[ {\frac{1}{2} + \frac{3}{{100}}} \right] + ... + \left[ {\frac{1}{2} + \frac{5}{{100}}} \right]$$ where $\left[ {A + B} \right]$ denotes the greatest integer function.
Solution: Taking the $L.C.M.$, we get
$$\begin{equation} \begin{aligned} = \left[ {\frac{{50 + 1}}{{100}}} \right] + \left[ {\frac{{50 + 2}}{{100}}} \right] + \left[ {\frac{{50 + 3}}{{100}}} \right] + ... + \left[ {\frac{{50 + 50}}{{100}}} \right] \\ = \left[ {\frac{{51}}{{100}}} \right] + \left[ {\frac{{52}}{{100}}} \right] + \left[ {\frac{{53}}{{100}}} \right] + ... + \left[ {\frac{{100}}{{100}}} \right] \\ = \left[ {0.51} \right] + \left[ {0.52} \right] + \left[ {0.53} \right] + ... + \left[ 1 \right] \\ = 0 + 0 + 0 + ... + 1 \\ = 1 \\\end{aligned} \end{equation} $$
