4.4 Transformation applied by fraction part function  (Page 2/2)

The lines drawn in step 3 is the graph of y=f({x}).

Fraction part function applied to the function

The form of transformation is depicted as :

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

The graph of y= f(x) is transformed in y={f(x)} by applying changes to the output of the function. Whatever be the function values, they will be changed to fraction values following definition of fraction part values as given earlier for few intervals. The values of y will lie in the interval [0,1).

Here, we need to recognize one important aspect of graph of real valued function. Consider a function value y=3. The function value such as y=3.3 shows a change in function value of 3.3-3=0.3. This change in function value depends on the integral part of y, which is 3. The change will be different at other integral part like 2 depending on the nature of function y = f(x). What it means that the nature of graph in the integral intervals of y have different set of fractional parts. In turn it means that when real values are converted to fractional part, resulting values represent different set of fraction parts, which is represented by the nature of graph segment between two consecutive integral intervals of y. Mathematically,

$$\left\{y\right\}=y-\left[y\right]$$

Clearly, {y} depends on y, but lies in the interval of y given by [0,1).

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 x-axis (horizontal lines) at integral values along y-axis to cover the graph of y=f(x).

2 : Identify segments of graph between two consecutive vertical intervals. Transfer these segments to y interval given by [0,1).

3 : Include end point corresponding to y=0 and exclude end point corresponding to y=1.

The lines drawn in step 3 is the graph of y={f(x)}.

Values assigned to fraction part function

The form of transformation is depicted as :

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

We need to evaluate this equation on the basis of assignment to the dependent expression variable. The value so evaluated is assigned to the FPF function {y}. We interpret assignment to {y} in accordance with the interpretation of equality of the FPF function to a value. In this case, we know that :

$$\left\{y\right\}=f\left(x\right);f\left(x\right)\in Z\Rightarrow \text{FPF can not be equated to integers. No solution.}$$

$$\left\{y\right\}=f\left(x\right);f\left(x\right)\notin Z\Rightarrow \text{y = Continuous interval of fraction values starting from f(x)}$$

Clearly, we need to neglect plot corresponding to integral values of f(x). On the other hand, there are multiple non-integral values of f(x) for a particular value of x corresponding to different intervals of unity along y. For example,

{-1.47}= {-0.47}= {0.53} = {1.53) = (2.53) = ….. = 0.53

Such is the case with other fractional values. It means that part of the graph of y=f(x) lying in y interval of [0,1) will be repeated in consecutive intervals of 1 along y-axis.

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 x-axis (horizontal lines) at integral values along y-axis to cover the graph of y=f(x).

2 : Identify part of the graph in y interval [0,1). Include end point corresponding to y=0 and exclude end point corresponding to y=1. Neglect other part of graph.

3 : Repeat part of graph identified in step 2 in other y intervals of unity along y-axis.

The lines drawn in step 3 is the graph of {y}= f(x).