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
5.6 Fractional Part 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 fractional part or decimal part of $x$ is denoted as, $$\left\{ x \right\}$$
Here, $$f(x) = \;\left\{ x \right\} = x - \left[ x \right]\;,\;\forall \;x \in R$$
For example: Let us assume the value of $x=4.9$ in a function $y = \left\{ x \right\}$. We can write $4.9$ as
$$\begin{equation} \begin{aligned} 4.9 = 4 + 0.9 \\ x = \left[ x \right] + \left\{ x \right\} \\ \Rightarrow \left\{ x \right\} = 0.9 \\ \therefore \left\{ x \right\} = x - \left[ x \right] \\\end{aligned} \end{equation} $$
To plot the graph, we can write the fractional part of $x$ as $$y = \left\{ x \right\} = x - \left[ x \right]$$
Now, let us divide the intervals of $x$ and find value of $y$.
| $y=x-[x]$ | $x$ |
| $x+3$ | $ - 3 \leqslant x < - 2$ |
| $x+2$ | $ - 2 \leqslant x < - 1$ |
| $x+1$ | $ - 1 \leqslant x < 0$ |
| $x$ | $0 \leqslant x < 1$ |
| $x-1$ | $ 1 \leqslant x < 2$ |
| $x-2$ | $2 \leqslant x < 3$ |
| $x-3$ | $ 3 \leqslant x < 4$ |
Using the values we get in the table, plot the graph in the corresponding intervals of $x$.
From the graph, we can conclude that, it is a periodic function with period $1$.
