Periodic Functions

 

For a given function \( f : A \rightarrow B \) defined by \( y = f(x) \), if there exists a non-zero real number \( T \) such that
\( \mathbf{f(x + T) = f(x)} \) for all \( x \in A \), then \( f \) is called a periodic function. The constant \( T \) is referred to as a period of the function, and the smallest positive value of \( T \) is called the fundamental period (or least period).

 

Example:

 

Let \( f \) and \( g \) be periodic functions defined on the set of real numbers. Given that \( f(x) = g(\displaystyle \frac{2x + 1}{3}) \) and the period of \( g(x) \) is 6, find the period of \( f(x) \).

Let \( T \) be the period of \( f(x) \).

\[ f(x + T) = f(x) \Rightarrow g(\displaystyle \frac{2(x + T) + 1}{3}) = g(\frac{2x + 1}{3}) \]

\[ \Rightarrow g(\displaystyle \frac{2x + 1 + 2T}{3}) = g(\frac{2x + 1}{3}) \]

\[ \Rightarrow g(\displaystyle \frac{2x + 1}{3} + \frac{2T}{3}) = g(\frac{2x + 1}{3}) \]

 

Since the period of the function \( g(x) \) is known to be 6, we set the horizontal shift equal to the period:

\[ \displaystyle \frac{2T}{3} = 6 \Rightarrow T = 9 \]

 

Example:

 

Find the fundamental period of the function \( f(x) = \cos (\displaystyle \frac{2x – 3}{5}) \).

Let \( T \) be a period of \( f(x) \).

\[ f(x + T) = f(x) \Rightarrow \cos (\displaystyle \frac{2(x + T) – 3}{5}) = \cos (\frac{2x – 3}{5}) \]
\[ \Rightarrow \cos (\displaystyle \frac{2x – 3 + 2T}{5}) = \cos (\frac{2x – 3}{5}) \]
\[ \Rightarrow \cos (\displaystyle \frac{2x – 3}{5} + \frac{2T}{5}) = \cos (\frac{2x – 3}{5}) \]
\[ \Rightarrow \displaystyle \frac{2x – 3}{5} + \frac{2T}{5} = \frac{2x – 3}{5} + 2k\pi \]
\[ \Rightarrow \displaystyle \frac{2T}{5} = 2k\pi \Rightarrow T = 5k\pi \]

 

Since \( k \in \mathbb{Z} \), the smallest positive value occurs at \( k = 1 \). Thus, the fundamental period is \( 5\pi \).

 

Periods of Trigonometric Functions:

 

1)

\[ f(x) = p + q \sin^n (ax + b) \]
\[ f(x) = p + q \cos^n (ax + b) \]
\[ f(x) = p + q \sec^n (ax + b) \]
\[ f(x) = p + q \csc^n (ax + b) \]

The fundamental periods for these functions are given by:

\[ \text{If } n \text{ is odd:} \quad \frac{2\pi}{|a|} \]

\[ \text{If } n \text{ is even:} \quad \displaystyle \frac{\pi}{|a|} \]

2)

\[ f(x) = p + q \tan^n (ax + b) \]
\[ f(x) = p + q \cot^n (ax + b) \]

The fundamental period for tangent and cotangent functions is always \( \displaystyle \frac{\pi}{|a|} \) for any integer value of n.

3) Let \( T_1 \) be the period of the trigonometric function \( g(x) \), and let \( T_2 \) be the period of the trigonometric function \( h(x) \):

a) For linear combinations of the form \( f(x) = g(x) \pm h(x) \), the fundamental period is given by the least common multiple: \( T = \text{LCM}(T_1, T_2) \).

b) For products of the form \( f(x) = g(x) \cdot h(x) \), the period is generally \( T = \text{LCM}(T_1, T_2) \). However, if the expression can be rewritten using product-to-sum identities as \( f(x) = g_1(x) \pm h_1(x) \) where \( g_1(x) \) has a period of \( t_1 \) and \( h_1(x) \) has a period of \( t_2 \), then the true fundamental period becomes \( t = \text{LCM}(t_1, t_2) \).

c) For quotients of the form \( f(x) = \displaystyle \frac{g(x)}{h(x)} \), the period can be checked via \( T = \text{LCM}(T_1, T_2) \). Note that the value obtained through LCM might not always be the absolute smallest fundamental period due to potential algebraic simplifications.

 

Examples:

 

• \( f(x) = 1 \, – \, 2 \sin^5 (3x \, – \, \displaystyle \frac{\pi}{3}) \quad \implies \quad T = \displaystyle \frac{2\pi}{3} \)

 

• \( f(x) = 3 \sec^4 (\displaystyle \frac{x}{2} + \pi) \quad \implies \quad T = \displaystyle \frac{\pi}{\frac{1}{2}} = 2\pi \)

 

• \( f(x) = 5 \tan (\, – \, 2x) \quad \implies \quad T = \displaystyle \frac{\pi}{|\, – \, 2|} = \frac{\pi}{2} \)

 

• \( f(x) = \cos^2 (\pi \, – \, \displaystyle \frac{2x}{3}) \quad \implies \quad T = \displaystyle \frac{\pi}{|\, – \, \frac{2}{3}|} = \frac{3\pi}{2} \)

 

• \( f(x) = 3 + \cot^3 (x + \pi) \quad \implies \quad T = \displaystyle \frac{\pi}{1} = \pi \)

 

QUESTION 67

 

What is the fundamental period of the function \( f(x) = 3 + 2 \tan \displaystyle \frac{x}{2} \, – \, \cot^2 (\, – \, x) + \sin^3 (\frac{x}{3} + \frac{\pi}{4}) \)?

 

\[ A) \ \pi \quad B) \ 2\pi \quad C) \ 4\pi \quad D) \ 6\pi \quad E) \ 8\pi \]

 

Solution:

 

The fundamental period of \( 2 \tan \displaystyle \frac{x}{2} \) is \( T_1 = \displaystyle \frac{\pi}{\frac{1}{2}} = 2\pi \).

The fundamental period of \( \cot^2 (\, – \, x) \) is \( T_2 = \displaystyle \frac{\pi}{|\, – \, 1|} = \pi \).

The fundamental period of \( \sin^3 (\displaystyle \frac{x}{3} + \frac{\pi}{4}) \) is \( T_3 = \displaystyle \frac{2\pi}{\frac{1}{3}} = 6\pi \).

Therefore, the fundamental period of the composite expression \( f(x) \) is found by taking the least common multiple of individual periods:

\[ T = \text{LCM}(T_1, T_2, T_3) = \text{LCM}(2\pi, \pi, 6\pi) = 6\pi \]

 

\( \textbf{Correct Answer: D} \)

 

QUESTION 68

 

What is the fundamental period of the function \( f(x) = \cos 5x \cdot \cos 3x \)?

 

\[ A) \ \pi \quad B) \ \frac{3\pi}{2} \quad C) \ 2\pi \quad D) \ 3\pi \quad E) \ \frac{7\pi}{2} \]

 

Solution:

 

Using the product-to-sum identity, we rewrite the function \( f(x) = \cos 5x \cdot \cos 3x \) as:

\[ f(x) = \displaystyle \frac{1}{2} (\cos 8x + \cos 2x) \]

The fundamental period of \( \cos 8x \) is \( T_1 = \displaystyle \frac{2\pi}{8} = \frac{\pi}{4} \).

The fundamental period of \( \cos 2x \) is \( T_2 = \displaystyle \frac{2\pi}{2} = \pi \).

Thus, the fundamental period of \( f(x) \) is:

\[ T = \text{LCM}(T_1, T_2) = \text{LCM}\left(\frac{\pi}{4}, \pi\right) = \pi \]

 

\( \textbf{Correct Answer: A} \)

 

QUESTION 69

 

What is the period of the function? \( f(x) = \displaystyle \frac{1 + \tan x}{\cos 2x} + \cot \frac{x}{2} \)

 

\[ A) \ 2\pi \quad B) \ 3\pi \quad C) \ 5\pi \quad D) \ 7\pi \quad E) \ 9\pi \]

 

Solution:

 

The fundamental period of the expression \( 1 + \tan x \) in the numerator is \( T_1 = \displaystyle \frac{\pi}{1} = \pi \).

The fundamental period of \( \cos 2x \) in the denominator is \( T_2 = \displaystyle \frac{2\pi}{2} = \pi \).

The fundamental period of the independent term \( \cot \displaystyle \frac{x}{2} \) is \( T_3 = \displaystyle \frac{\pi}{\frac{1}{2}} = 2\pi \).

Taking the least common multiple to determine the net period of \( f(x) \):

\[ T = \text{LCM}(T_1, T_2, T_3) = \text{LCM}(\pi, \pi, 2\pi) = 2\pi \]

 

\( \textbf{Correct Answer: A} \)