Problem 123
Question
Which of the following functions have period \(2 ?\) (A) \(\\{x\\}+\cos \pi x\) (B) \(\tan \left(\frac{\pi}{2}[x]\right)\) (C) \(\sin x+\\{x\\}\) (D) \(\sin (\cos x)\)
Step-by-Step Solution
Verified Answer
None of the functions have period 2.
1Step 1: Analyze Function A
We are given the function \( \{x\} + \cos \pi x \), where \( \{x\} \) is the fractional part of \( x \). The fractional part has period 1, and \( \cos \pi x \) has period 2. The period of a sum of functions is the least common period when defined. \( \{x\} + \cos \pi x \) has the smallest period due to \( \{x\} \). Thus, this function does not have period 2.
2Step 2: Analyze Function B
Consider the function \( \tan \left(\frac{\pi}{2}[x]\right) \), where \( [x] \) is the greatest integer function. This function involves the tangent, which becomes undefined at \( \frac{\pi}{2} \) increments when \( [x] \) becomes an integer, i.e., at every integer multiple of \( \pi \). However, \( [x] \) itself shows step-like behavior, which is not periodic at 2. Therefore, this function does not have a period 2.
3Step 3: Analyze Function C
The function \( \sin x + \{x\} \) involves both \( \sin x \), which has period \( 2\pi \), and \( \{x\} \), which has period 1. Since \( \{x\} \) has the shortest period, the combination will repeat every 1, but neither 1 nor the combined function can match the condition for period 2. Hence, this function does not have period 2.
4Step 4: Analyze Function D
Now consider \( \sin(\cos x) \). Here, \( \cos x \) is periodic with period \( 2\pi \), and since \( \sin \) is periodic with period \( 2\pi \), \( \sin(\cos x) \) is not expected to have a period of 2. A detailed analysis replacing various interval transformations quickly reveals no simplification towards mutual period 2, failing period 2.
5Step 5: Conclusion
None of the functions presented have period 2 as discussed with period interactions and transformations resulting in harmonic analysis verifying that no synchronized period equal to 2 emerges.
Key Concepts
PeriodicityFractional Part FunctionGreatest Integer Function
Periodicity
Periodicity is a key concept in mathematics, especially in the study of functions like trigonometric functions. It refers to the property of a function to repeat values at regular intervals, known as periods. For example, the function \( \sin x \) has a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) units along the x-axis.
One important aspect is the period of a sum of functions. When combining two periodic functions, such as \( \{x\} + \cos \pi x \), their combined period is determined by the least common multiple of their individual periods. However, it's not always straightforward, as some functions might limit the overall periodicity due to their smaller periods. In this example, while \( \cos \pi x \) has a period of 2, the fractional part function \( \{x\} \) has a period of 1, thus governing the period rule and not allowing the function \( \{x\} + \cos \pi x \) to have a period of 2.
Understanding periodicity helps in analyzing and predicting the behavior of functions over time, especially when dealing with complex combinations.
One important aspect is the period of a sum of functions. When combining two periodic functions, such as \( \{x\} + \cos \pi x \), their combined period is determined by the least common multiple of their individual periods. However, it's not always straightforward, as some functions might limit the overall periodicity due to their smaller periods. In this example, while \( \cos \pi x \) has a period of 2, the fractional part function \( \{x\} \) has a period of 1, thus governing the period rule and not allowing the function \( \{x\} + \cos \pi x \) to have a period of 2.
Understanding periodicity helps in analyzing and predicting the behavior of functions over time, especially when dealing with complex combinations.
Fractional Part Function
The fractional part function, denoted \( \{x\} \), is a fascinating mathematical concept. It represents the difference between a real number and its greatest integer less than or equal to it. For instance, \( \{4.75\} = 0.75 \). This fractional component captures any remaining part of the number that isn't an integer.
This function is unique as it naturally has a period of 1. What this means is that for any integer \( n \), the function \( \{x + n\} \) equals \( \{x\} \). This property makes \( \{x\} \) quite useful in working with periodic functions because it ensures the repetition of values at every unit increment.
When combined with other functions, like in \( \sin x + \{x\} \), it can have significant implications on the overall periodicity. Despite the periodicity of the sine function being \( 2\pi \), the fractional part function's periodicity tends to dominate due to its shorter period, typically resulting in much more frequent repetitions.
This function is unique as it naturally has a period of 1. What this means is that for any integer \( n \), the function \( \{x + n\} \) equals \( \{x\} \). This property makes \( \{x\} \) quite useful in working with periodic functions because it ensures the repetition of values at every unit increment.
When combined with other functions, like in \( \sin x + \{x\} \), it can have significant implications on the overall periodicity. Despite the periodicity of the sine function being \( 2\pi \), the fractional part function's periodicity tends to dominate due to its shorter period, typically resulting in much more frequent repetitions.
Greatest Integer Function
The greatest integer function, often denoted \( [x] \), is also known as the floor function. This function maps a real number to the largest previous integer. For example, \([3.14] = 3\) and \([-2.78] = -3\).
This function is unique because its graph appears as a series of constant "steps," jumping to the next integer as the input reaches integer values. These steps mean that \( [x] \) is not periodic in the traditional sense, as its outputs do not repeat in regular increments.
In complex expressions, such as \( \tan \left(\frac{\pi}{2}[x]\right) \), the greatest integer function can cause sudden changes in the function's behavior. Here, each integer multiple of \( \pi/2 \) can make the tangent function undefined, introducing complications in identifying a regular period such as 2. Understanding how \( [x] \) pairs with other operations is vital in predicting function outcomes.
This function is unique because its graph appears as a series of constant "steps," jumping to the next integer as the input reaches integer values. These steps mean that \( [x] \) is not periodic in the traditional sense, as its outputs do not repeat in regular increments.
In complex expressions, such as \( \tan \left(\frac{\pi}{2}[x]\right) \), the greatest integer function can cause sudden changes in the function's behavior. Here, each integer multiple of \( \pi/2 \) can make the tangent function undefined, introducing complications in identifying a regular period such as 2. Understanding how \( [x] \) pairs with other operations is vital in predicting function outcomes.
Other exercises in this chapter
Problem 118
If \(f\) is an even function defined on the interval \([-5,5]\), then the real values of \(x\) satisfying the equation $$ f(x)=f\left(\frac{x+1}{x+2}\right) \te
View solution Problem 119
The distinct linear function which maps \([-1,1]\) onto \([0,2]\) is (A) \(x-1\) (B) \(x+1\) (C) \(-x+1\) (D) \(-x-1\)
View solution Problem 124
The values of \(x\) for which the domain of definition of the function, \(f(x)=\frac{1}{[|x-1|]+|7-x|-6}\), where [.] denotes the greatest integer part, is not
View solution Problem 126
If \(f: R \rightarrow R\) be defined by \(f(x)=\frac{e^{x}-e^{-x}}{2}\), then (A) \(f\) is one-one (B) \(f\) is onto (C) \(f^{-1}(x)=\log \left(x-\sqrt{x^{2}+1}
View solution