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Transformation applied by fraction part function

Module by: Sunil Kumar Singh. E-mail the author

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.

Figure 1: FPF is a periodic function with period 1.
Fraction part function
 Fraction part function  (f1.gif)

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.) :

{ x } = x + 2 ; - 2 x < - 1 { x } = x + 2 ; - 2 x < - 1 { x } = x + 1 ; - 1 x < 0 { x } = x + 1 ; - 1 x < 0 { x } = x ; 0 x < 1 { x } = x ; 0 x < 1 { x } = x 1 ; 1 x < 2 { x } = x 1 ; 1 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 x y = f { x } y = f x y = f { x }

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

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

Example 1

Problem : Draw the graph of sin{x}.

Solution : Following the construction steps, graph of y=sin{x} is drawn by transforming y = sinx as shown here.

Figure 2: Repeat the part of the graph identified between 0 and 1 to other intervals of x.
Graph of y=sin{x}
 Graph of y=sin{x}  (f2.gif)

Example 2

Problem : Draw the graph of y = e x e [ x ] y = e x e [ x ] .

Solution : Rearranging, we have :

y = e x e [ x ] = e x [ x ] = e { x } y = e x e [ x ] = e x [ x ] = e { x }

Following the construction steps, graph of y = e { x } y = e { x } is drawn by transforming y = e x y = e x as shown here.

Figure 3: Repeat the part of the graph identified between 0 and 1 to other intervals of x.
Transformation of exponential graph
 Transformation of exponential graph
 (f3.gif)

Fraction part function applied to the function

The form of transformation is depicted as :

y = f x y = { f x } y = f x y = { f x }

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,

{ y } = y [ y ] { y } = y [ y ]

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)}.

Example 3

Problem : Draw the graph of { log e x } { log e x } .

Solution : Following the construction steps, graph of y = { log e x } y = { log e x } is drawn as shown here.

Figure 4: Transfer part of the graph identified in unit y interval [0,1).
Transformation of sine graph
 Transformation of sine graph
  (f4.gif)

Example 4

Problem : Draw the graph of {2sinx}.

Solution : Following the construction steps, graph of y={2sinx} is drawn by transforming y= 2sinx as shown here.

Figure 5: Transfer part of the graph identified in unit y-interval [0,1).
Graph of y={sinx}
 Graph of y={sinx}  (f5.gif)

Note two individual solid circles on x-axis. They have been enclosed in squares for emphasis. We should analyze their existence while constructing the graph.

Values assigned to fraction part function

The form of transformation is depicted as :

y = f x { y } = f x y = f x { y } = f x

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 :

{ y } = f x ; f x Z FPF can not be equated to integers. No solution. { y } = f x ; f x Z FPF can not be equated to integers. No solution.

{ y } = f x ; f x Z y = Continuous interval of fraction values starting from f(x) { y } = f x ; f x Z 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).

Example 5

Problem : Draw graph of {y}= sinx; x∈[-2π,2π].

Solution : Following construction steps, graph of {y}= sinx is drawn by transforming y= sinx as shown here.

Figure 6: Identify part of the graph in y interval [0,1). Repeat part of graph so identified in other y intervals of unity
Graph of y=sin{x}
 Graph of y=sin{x}  (f6.gif)

Example 6

Problem : Draw graph of { y } = e x { y } = e x .

Solution : Following construction steps, graph of { y } = e x { y } = e x is drawn by transforming y = e x y = e x as shown here.

Figure 7: Identify part of the graph in y interval [0,1). Repeat part of graph so identified in other y intervals of unity
Graph of y=sin{x}
 Graph of y=sin{x}  (f7.gif)

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