5.2 Periodic functions  (Page 3/5)

From the data in table, we infer that difference as given by “x – [x]” is periodic with a period of “1”. Note that function value repeats for an increment of "1" in the value of "x". We, now, proceed to prove this analytically. Here,

$$f\left(x\right)=x-\left[x\right]$$

$$\Rightarrow f\left(x+T\right)=x+T-[x+T]$$

Let us assume that the given function is indeed a periodic function. Then by definition,

$$f\left(x+T\right)=f\left(x\right)$$

$$\Rightarrow x+T-[x+T]=x-\left[x\right]$$

$$\Rightarrow T=[x+T]-\left[x\right]$$

$$\Rightarrow T\in Z$$

Clearly, “T” is an integer as both greatest integer functions return integers. There exists T>0, which satisfies the equation f(x+T) = f(x). The least positive integer is “1”. Hence, period of the function is “1”.

Arithmetic operations and periodicity

A periodic function can be modified by arithmetic operations on independent variable of the function or function itself. The arithmetic operations involved here are addition, subtraction, multiplication, division and negation. We have studied (read module titled transformation of graphs) these operations and seen that there are different effects on the graph of core function due to these operations. Arithmetic operations on independent variable change input to the function and the graph of core function is transformed horizontally (along x-axis). On the other hand, operations on the function itself change output and the graph of core function is transformed vertically (along. y-axis). The combined input/output arithmetic operations related to function are symbolically represented as :

$$af(bx+c)+d;\mathrm{a,b,c,d}\in Z$$

Important thing to understand here is that periodicity is defined in terms of independent variable, x. A periodic function repeats a set of its values after regular interval of independent variable i.e. x., Clearly, periodicity of a periodic function is not affected by transformations in vertical direction. Hence, arithmetic operations with function involving constants “a” and “d” do not affect periodicity of a periodic function.

Not all arithmetic operations on independent variable will change or affect periodicity. Shifting of core graph due to addition or subtraction results in shifting of the graph as a whole either to the left or right. This operation does not change size and shape of the graph. Thus, addition and subtraction operation involving constant “c” does not affect periodicity of a function. Negation of independent variable, when “b” is negative, results in flipping of the graph without any change in size and shape of the graph. As such, negation of independent variable does not change periodicity either.

It is only the multiplication or division of independent variable x by a positive constant, “b” greater than 1, result in change in size with respect to origin in horizontal direction. The graph shrinks horizontally when independent variable is multiplied by positive constant greater than 1 by the factor which is equal to the multiplier. This means periodicity of graph decreases by the same factor i.e.|b|. The graph stretches horizontally when independent variable is divided by positive constant greater than 1 by the factor which is equal to the divisor. This means periodicity of graph increases by the same factor i.e.|b|. We combine these two observations by saying that period of graph decrease by a factor |b|. Note that magnitude of constant “b” more than 1 represents multiplication and less than represents division.