Problem 79
Question
$$ \begin{aligned} f(x) &=[6 x+7]+\cos \pi x-6 x \\ &=[6 x]+7+\cos \pi x-6 x \\ &=7+\cos \pi x-\\{6 x\\} \end{aligned} $$ \(\\{6 x\\}\) has period \(1 / 6\) and \(\cos \pi x\) has period 2. Then, Period of \(f(x)=L C M\) of 2 and \(1 / 6=2\) Hence, the period is 2 .
Step-by-Step Solution
Verified Answer
The period of the function is 2.
1Step 1: Identify the Period of Each Function
The function consists of two parts: \(\{6x\}\) and \(\cos \pi x\). Identify the individual periods:- The fractional part \(\{6x\}\) has a period of \(\frac{1}{6}\) since its period is the reciprocal of the coefficient of \(x\).- The cosine function \(\cos \pi x\) has a period of 2 because the period of \(\cos\) is \(\frac{2\pi}{\pi} = 2\).
2Step 2: Calculate the Least Common Multiple (LCM)
Calculate the least common multiple (LCM) of the periods obtained in step 1:- LCM of \(2\) and \(\frac{1}{6}\) is the smallest number that both periods can divide evenly into.- Convert \(2\) to \(\frac{12}{6}\) to compare.- Thus, LCM is \(2\) since it's the larger period that encapsulates full wavelengths of both functions.
3Step 3: Confirm the Overall Period
The overall period of the function \(f(x)\) is determined by the LCM calculated:- Since the LCM is \(2\), the overall period of \(f(x)\) is \(2\).- This means that the function \(f(x)\) will repeat every \(2\) units along the x-axis.
Key Concepts
Fractional Part FunctionCosine FunctionLeast Common Multiple
Fractional Part Function
The fractional part function, often denoted as \(\{x\}\), refers to the part of a number that comes after the decimal point. Consider a number, say \(x = 2.75\). The integer part is 2, and the fractional part is \(0.75\). The fractional part function extracts the \(0.75\) from \(2.75\). This function is crucial in mathematical analysis and has periodic properties.
For any real number \(x\), the function \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the floor or integer part of \(x\).
For any real number \(x\), the function \(\{x\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the floor or integer part of \(x\).
- Example: For \(x = 4.36\), \(\{x\} = 0.36\).
- The period of the fractional part function \(\{kx\}\) (where \(k\) is a constant) is \(\frac{1}{k}\). This is because shifting \(x\) by \(\frac{1}{k}\) results in a cycle completion.
- In our exercise, \(\{6x\}\) means every \(\frac{1}{6}\) the value repeats, highlighting the essence of periodicity in fractional functions.
Cosine Function
The cosine function, denoted as \(\cos(x)\), is a fundamental trigonometric function that describes the horizontal coordinate of a point on the unit circle as it travels through an angle \(x\). It's known for its wave-like pattern and periodic nature.
The basic form of the cosine function is \(\cos(\theta)\), which has a period of \(2\pi\) radians or 360 degrees. This means the cosine wave repeats itself every \(2\pi\) radians.
The basic form of the cosine function is \(\cos(\theta)\), which has a period of \(2\pi\) radians or 360 degrees. This means the cosine wave repeats itself every \(2\pi\) radians.
- When transformation is applied, like in \(\cos(\pi x)\), the period changes because it modifies the angle's frequency.
- Specifically, the period for \(\cos(\pi x)\) is 2 since \(\frac{2\pi}{\pi} = 2\).
- Understanding transformations like these helps in predicting and analyzing oscillating patterns found in numerous scientific and engineering problems.
Least Common Multiple
The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. It's a key concept when dealing with the periodic functions as it assists in determining the overall period for a function composed of multiple periodic functions.
For example, in our exercise, we deal with two periods:
To find the period of the combined function \(f(x)\), we need the LCM of \(\frac{1}{6}\) and 2. The trick here is to convert both values to have a common denominator for an easy comparison. So, \(2\) becomes \(\frac{12}{6}\). The smallest number that can accommodate both is 2. Thus, the LCM or the overall period of \(f(x)\) is 2.
For example, in our exercise, we deal with two periods:
- The period of the fractional part function\(\{6x\}\) is \(\frac{1}{6}\).
- The period of \(\cos(\pi x)\) is 2.
To find the period of the combined function \(f(x)\), we need the LCM of \(\frac{1}{6}\) and 2. The trick here is to convert both values to have a common denominator for an easy comparison. So, \(2\) becomes \(\frac{12}{6}\). The smallest number that can accommodate both is 2. Thus, the LCM or the overall period of \(f(x)\) is 2.
- LCM is invaluable in syncing cycles of different periodic functions, ensuring a smooth, unified repetition period.
Other exercises in this chapter
Problem 74
$$ \begin{aligned} f(-x)=\frac{\cos (-x)}{\left[-\frac{2 x}{\pi}\right]+\frac{1}{2}} &=\frac{\cos x}{-1-\left[\frac{2 x}{\pi}\right]+\frac{1}{2}} \quad \begin{a
View solution Problem 75
For odd function, $$ \begin{aligned} f(x) &=-f(-x) \\ &=- \begin{cases}\sin (-x)+\cos (-x) & 0
View solution Problem 91
\(f(x)\) is defined for \(x \in(0,1)\) So, \(f\left(e^{x}\right)+f(\ln |x|)\) is defined for \(0
View solution Problem 94
\(y=2^{x(x-1)}\) or \(x^{2}-x-\log _{2} y=0\) $$ \text { or } \quad x=\frac{1}{2}\left(1 \pm \sqrt{1+4 \log _{2} y}\right) $$ Since \(x \in[1, \infty)\), we cho
View solution