4.4 Transformation applied by fraction part function

Drawings of graphs resulting from transformation applied by fraction part function (FPF) follow the same reasoning and steps as deliberated for modulus and greatest integer function. Before, we proceed to draw graphs for different function forms, we need to recapitulate the graph of fraction part function (FPF) and also infer thereupon few of the values of FPF around zero.

The values of FPF around zero are (we can write these expressions by observing graph. A bit of practice to write down these intervals helps.) :

$$\left\{x\right\}=x+2;-2\le x<-1$$ $$\left\{x\right\}=x+1;-1\le x<0$$ $$\left\{x\right\}=x;0\le x<1$$ $$\left\{x\right\}=x-1;1\le x<2$$

Important point to note is that lower integer is included, but higher integer is excluded in the intervals of unity in which FPF is defined. The graph segment in the interval [0,1) is y=x i.e. identity function. We obtain expression of function in right intervals (positive value intervals of x) by shifting identity function towards right by 1 successively and in left intervals (negative value intervals of x) by shifting identity function towards left by 1 successively. The values of FPF are continuous real values which is equal to or greater than zero but less than 1. These function values are repeated in each of intervals of unity along x-axis. Thus, FPF is a periodic function with period of 1. Domain of FPF is R and range is [0,1). Further, FPF is related to real number as x=[x]+{x}.

A function like y=f(x) has different elements. We can apply FPF to these elements of the function. There are following different possibilities :

1 : y = f({x})

2 : y ={f(x)}

3 : {y} = f(x)

Fraction part operator applied to the argument

The form of transformation is depicted as :

$$y=f\left(x\right)\Rightarrow y=f\left(\left\{x\right\}\right)$$

The graph of y=f(x) is transformed in y=f({x}) by virtue of changes in the argument values. The independent variable is subjected to fraction part operator. This changes the normal real value input to function. Instead of real numbers, independent variable to function is rendered to be fractions irrespective of values of x. A value like x = - 2.3 is passed to the function as 0.7 in the interval [0,1).

Clearly, real values of “x” are truncated to fraction values in all intervals. It means that same set of values of the function y=f(|x|) corresponding to interval of x defined by [0,1] will repeat in other intervals along x-axis. The FPF is a periodic function with a period of 1. Taking advantage of this fact, we obtain graph of y=f({x}) by repeating part of graph for x in [0,1) to other intervals along x-axis. Clearly, transformed function y=f({x}) is periodic with a period of 1.

From the point of construction of the graph of y=f({x}), we need to modify the graph of y=f(x) as :

1 : Draw lines parallel to y-axis (vertical lines) at integral values along x-axis to cover the graph of y=f(x).

2 : Identify part of the graph for values of x in [0,1). Include end point corresponding to x=0 and exclude end point corresponding to x=1.

3 : Repeat the part of the graph identified in step 2 for other intervals of x