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Magazine Chapter 11: Problem 4
Show that \(f=\) const is periodic with any period but has a fundamental period.
Short Answer
Expert verified
A constant function is periodic with any period but does not have a fundamental period.
Step by step solution
01
Understand the Definition of a Constant Function
A constant function is defined as a function that returns the same value for any input. Mathematically, it is written as \(f(x) = c\), where \(c\) is a constant and does not depend on \(x\).
02
Define Periodic Function
A function \(f(x)\) is said to be periodic with period \(T\), if for all \(x\) in the domain of \(f\), \(f(x + T) = f(x)\).
03
Show Any Period Works
For \(f(x) = c\), we have \(f(x + T) = c = f(x)\) for any value of \(T > 0\). This is because the output of the function is always \(c\) regardless of the input or the period chosen.
04
Define Fundamental Period
The fundamental period of a function is the smallest positive period \(T_0\) for which \(f(x + T_0) = f(x)\) holds true for all \(x\).
05
Determine Fundamental Period for Constant Function
Since \(f(x) = c\) holds for any \(T\), there is no smallest positive \(T > 0\) that satisfies the period condition other than the trivial case of zero, i.e., there is no meaningful smallest positive period for \(f(x) = c\). Therefore, a constant function does not have a fundamental period.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Constant Function
A constant function is one of the simplest types of mathematical functions. It is defined by its unique characteristic: it always returns the same value, regardless of the input. This can be expressed in mathematical terms as \( f(x) = c \). Here, \( c \) represents a constant number that doesn't change with different values of \( x \).
This simplicity makes it quite unique.
Constant functions are linear and can be found as horizontal lines on a graph.
This simplicity makes it quite unique.
Constant functions are linear and can be found as horizontal lines on a graph.
- They have zero slope.
- They do not cross the x-axis unless the constant value itself is zero, in which case, it lies along the x-axis.
Fundamental Period
The concept of the fundamental period is essential when discussing periodic functions. The fundamental period of a function is the smallest positive value \( T_0 \) such that the function repeats itself: \( f(x + T_0) = f(x) \) for all \( x \).
It acts like the "duration" it takes for the function's behavior to loop back to its starting point.
In the case of a constant function, however, each value \( f(x) = c \) is equal to \( f(x + T) = c \) for any \( T > 0 \).
This means that there is no smallest positive period because it's essentially "always" repeating, in a sense being perpetually constant.
It acts like the "duration" it takes for the function's behavior to loop back to its starting point.
In the case of a constant function, however, each value \( f(x) = c \) is equal to \( f(x + T) = c \) for any \( T > 0 \).
This means that there is no smallest positive period because it's essentially "always" repeating, in a sense being perpetually constant.
- The fundamental period is usually a crucial feature of more complex periodic functions like sine or cosine.
- But for a constant function, this concept doesn't hold the same weight.
Definition of Periodic Function
Periodic functions are functions that repeat their values in regular intervals or periods.
By definition, a function \( f(x) \) is periodic with period \( T \) if, for every \( x \) in the function's domain, it holds true that \( f(x + T) = f(x) \).
This characteristic is what allows periodic functions to exhibit repeating patterns over consistent intervals.
By definition, a function \( f(x) \) is periodic with period \( T \) if, for every \( x \) in the function's domain, it holds true that \( f(x + T) = f(x) \).
This characteristic is what allows periodic functions to exhibit repeating patterns over consistent intervals.
- Examples of periodic functions include the sine and cosine functions.
- The key feature is having a clearly defined period \( T \).
Mathematical Function Properties
Understanding the properties of mathematical functions helps in categorizing them and predicting their behavior.
Here are some key properties of functions:
Here are some key properties of functions:
- Continuity: Functions are continuous if there are no breaks, jumps, or holes in their graphs.
- Linearity: Functions are linear if they can be graphed as straight lines. Constant functions fall under this category, but specifically as horizontal lines.
- Periodic Behavior: As seen, periodic functions repeat values over intervals, but for constant functions, this "periodicity" is trivial as they maintain a constant output.
