Problem 116
Question
$$ f(3 x+2)+f(3 x+29)=0 $$ Replacing \(x\) by \(x+9\), we get $$ f(3(x+9)+2)+f(3(x+9)+29)=0 $$ or \(f(3 x+29)+f(3 x+56)=0\) From (1) and (2), we get $$ f(3 x+2)=f(3 x+56) $$ or \(f(3 x+2)=f(3(x+18)+2)\) Therefore, \(f(x)\) is periodic with period 54 .
Step-by-Step Solution
Verified Answer
The function \( f(x) \) is periodic with period 54.
1Step 1: Understanding the Original Equation
We start by understanding the given equation: \( f(3x+2) + f(3x+29) = 0 \). This equation is crucial as it states that for a particular \( x \), the sum of \( f \) values at two different points is zero.
2Step 2: Substituting \( x+9 \) into the Equation
Next, we substitute \( x+9 \) into the original equation. This gives us another equation: \( f(3(x+9)+2) + f(3(x+9)+29) = 0 \) which simplifies to \( f(3x+29) + f(3x+56) = 0 \). This is a transformation of the original equation for a shifted \( x \).
3Step 3: Comparing the Two Equations
Now, we compare the two equations obtained: \( f(3x+2) + f(3x+29) = 0 \) and \( f(3x+29) + f(3x+56) = 0 \). From these, we see that \( f(3x+2) = f(3x+56) \). This implies that the function \( f \) has the same value at points separated by 54 units.
4Step 4: Deriving the Periodicity
Identifying \( f(3x+2) = f(3x+56) \), we recognize a pattern of periodicity. The points are separated by 54, denoting that the function \( f(x) \) has a period of 54.
Key Concepts
Function PeriodicityJEE Calculus ProblemsShifting Transformations
Function Periodicity
Periodicity in functions is a crucial concept in calculus and various applications.When a function is said to be periodic, it means that its values repeat at consistent intervals or periods.
These functions are essential because they model many natural and mechanical systems that exhibit cyclical patterns.In the exercise we've seen, by using the equations provided, the periodic nature of the function was determined.Let's quickly review:- We saw that for the function given, the equation was transformed with the variable substitution method.- As a result, it became apparent that the function values repeated every 54 units.Periodic functions simplify calculus and problem-solving in scenarios where patterns repeat.If a function is periodic with period \( T \), it means:\[ f(x) = f(x + T) \]This kind of function can be easily analyzed and predicted across its entire domain by understanding just one period.
These functions are essential because they model many natural and mechanical systems that exhibit cyclical patterns.In the exercise we've seen, by using the equations provided, the periodic nature of the function was determined.Let's quickly review:- We saw that for the function given, the equation was transformed with the variable substitution method.- As a result, it became apparent that the function values repeated every 54 units.Periodic functions simplify calculus and problem-solving in scenarios where patterns repeat.If a function is periodic with period \( T \), it means:\[ f(x) = f(x + T) \]This kind of function can be easily analyzed and predicted across its entire domain by understanding just one period.
JEE Calculus Problems
JEE (Joint Entrance Examination) calculus problems often incorporate periodic functions to test understanding of concepts such as transformations and cyclic properties.
These problems might seem daunting at first, but understanding the basic principles of periodicity and transformations can make them much more manageable.
Here’s how periodicity surfaces in JEE calculus problems:
- **Pattern recognition**: Identifying periodic patterns can lead to simpler solutions.
- **Period manipulation**: Shifts and substitutions are frequently used to reveal inherent periodicity.
- **Proof**: You may be asked to prove that a function is periodic, making it essential to understand how substitutions and comparisons work.
Let's cover a few helpful tips for tackling these problems:
- Always look for transformations that could simplify the equation.
- Break down complex expressions to see if there are terms repeating after a certain interval.
- Verify if substituting specific values reveals a periodic pattern.
With practice, recognizing and utilizing the periodic nature of functions in JEE will become a valuable skill.
Shifting Transformations
Shifting transformations are mathematical techniques used to simplify function equations, especially in calculus.When dealing with periodic functions, these transformations can make it easier to spot recurring patterns.
In our example exercise, shifting was used by replacing \( x \) with \( x+9 \) to transform the original equation.This substitution revealed the periodicity by comparing and simplifying the new and original equations.
Key points to remember about shifting transformations:- **Horizontal Shifts**: Involves adding/subtracting a constant to/from \( x \). This changes the position of the function graph horizontally.- **Vertical Shifts**: Adding/subtracting a constant to/from the function's equation, affects the graph's vertical position.- **Purpose**: Useful for finding links between function values and periods by aligning similar terms.Employing shifting transformations can help you uncover hidden properties of functions, making problems like those often found in calculus more approachable.
In our example exercise, shifting was used by replacing \( x \) with \( x+9 \) to transform the original equation.This substitution revealed the periodicity by comparing and simplifying the new and original equations.
Key points to remember about shifting transformations:- **Horizontal Shifts**: Involves adding/subtracting a constant to/from \( x \). This changes the position of the function graph horizontally.- **Vertical Shifts**: Adding/subtracting a constant to/from the function's equation, affects the graph's vertical position.- **Purpose**: Useful for finding links between function values and periods by aligning similar terms.Employing shifting transformations can help you uncover hidden properties of functions, making problems like those often found in calculus more approachable.
Other exercises in this chapter
Problem 106
\(f(x)=x^{n}+1\) or \(\quad f(3)=3^{n}+1=28\) or \(\quad 3^{n}=27\) \(\therefore \quad n=3\) or \(f(4)=4^{3}+1=65\)
View solution Problem 114
$$ \left(x+\frac{1}{2}\right)+f\left(x-\frac{1}{2}\right)=f(x) $$ Replacing \(x\) by \(\left(x+\frac{1}{2}\right)\), we get $$ f(x+1)+f(x)=f\left(x+\frac{1}{2}\
View solution Problem 117
From the given data, $$ f(1-x)=f(1+x) $$ and \(f(2-x)=f(2+x)\) \(\ln (2)\), replacing \(x\) by \(1+x\), we have $$ f(1-x)=f(3+x) $$ or \(f(1+x)=f(3+x)\) or \(f(
View solution Problem 119
\(|x-2|+a=\pm 4\) or \(|x-2|=\pm 4-a\) For four real roots, \(4-a>0\) and \(-4-a>0\) or \(a \in(-\infty,-4)\)
View solution