How to Find the Period of a Function: A Comprehensive Guide

Understanding the periodicity of functions is a fundamental concept in mathematics, with applications ranging from physics and engineering to computer science and signal processing. A periodic function is simply one that repeats its values at regular intervals. Identifying this repeating interval, known as the period, is crucial for analyzing and predicting the behavior of these functions. This comprehensive guide will delve into the concept of periodicity, explain how to find the period of various types of functions, and provide numerous examples to solidify your understanding.

What is a Periodic Function?

A function f(x) is considered periodic if there exists a positive number P such that:

f(x + P) = f(x)

for all values of x in the domain of the function. This number P is called the period of the function. It’s the smallest positive value for which this condition holds true. Geometrically, this means that the graph of the function repeats itself after every interval of length P.

Think of a sine wave; it oscillates up and down, repeating the same pattern over and over. The distance along the x-axis it takes for one complete cycle of this pattern to occur is its period.

It’s crucial to note that if f(x+P) = f(x), it also implies that f(x+2P) = f(x), f(x+3P) = f(x), and so on. However, the fundamental or smallest period is what we typically refer to when discussing the period of a function.

Types of Functions and their Periods

Before diving into the methods, it’s helpful to know the common types of functions you’ll encounter and their typical periods:

• Trigonometric Functions:
  • Sine (sin x): Period = 2π
  • Cosine (cos x): Period = 2π
  • Tangent (tan x): Period = π
  • Cosecant (csc x): Period = 2π
  • Secant (sec x): Period = 2π
  • Cotangent (cot x): Period = π
• Linear Functions (y = ax + b): Generally, linear functions are not periodic unless the slope ‘a’ is zero, in which case it’s just a constant function and doesn’t technically have a period in the way we are discussing.
• Polynomial Functions (y = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀): Polynomial functions (other than constants) are typically not periodic.
• Exponential Functions (y = a^x): Exponential functions are generally not periodic.
• Absolute value functions (e.g., y = |sin x|): Can be periodic depending on the inner function (e.g. |sin x| has period π)
• Combined functions : The period can be found by using LCM of the period of respective functions in some cases.

Methods to Find the Period of a Function

The method to find the period of a function often depends on the type of function you’re working with. Here are common approaches:

1. Using the Definition (Substitution Method)

This is the most fundamental approach and works for many basic functions. Here’s how to use it:

1. Start with the definition: f(x + P) = f(x)
2. Substitute: Replace x with (x + P) in the function’s expression.
3. Solve for P: Simplify the equation and solve for the smallest positive value of P that makes the equation true for all x.

Example 1: Finding the period of f(x) = sin(x)

1. Definition: sin(x + P) = sin(x)
2. Trigonometric Identity: Recall the sine addition formula: sin(a + b) = sin(a)cos(b) + cos(a)sin(b), applying this we get sin(x+P) = sin(x)cos(P) + cos(x)sin(P)
3. Substitution: sin(x) = sin(x)cos(P) + cos(x)sin(P). For this to be true, we must have that cos(P) =1 and sin(P) = 0.
4. Solving for P: The smallest positive value for P that satisfies these conditions is P = 2π. Thus, the period of sin(x) is 2π.

Example 2: Finding the period of f(x) = cos(2x)

1. Definition: cos(2(x + P)) = cos(2x)
2. Simplification: cos(2x + 2P) = cos(2x)
3. Recognize the period of cos(x): We know cos(θ) is 2π periodic so the argument difference here (2x + 2P) – 2x should be multiple of 2π
4. Solving for P: For the argument to go through one cycle we must have 2P = 2π and hence P = π

Example 3: Finding the period of f(x) = tan(x/3)

1. Definition: tan((x + P)/3) = tan(x/3)
2. Simplification: tan((x/3)+ (P/3)) = tan(x/3)
3. Recognize the period of tan(x): We know that tan(θ) has a period of π so we have P/3 = π
4. Solving for P: P= 3π

Example 4: Finding the period of f(x) = |sin(x)|

1. Definition: |sin(x + P)| = |sin(x)|
2. Graphing : If you plot a graph you can quickly recognize that the period is π, However, let’s try by formula.
3. Using the definition : If the period was 2π, we’d have, |sin(x+2π)| = |sin(x)| and this is true. But, we need to find the smallest period.
4. Using symmetry : We know that |sin(π + x)| = |-sin(x)| = |sin(x)| and |sin(π – x)| = |sin(x)|. So the function has period π.

2. Period of Transformed Trigonometric Functions

Trigonometric functions are often modified by transformations. Understanding how these affect the period is essential. Here are some rules:

• f(x) = A sin(Bx + C) + D:
  • A affects the amplitude (height) of the wave.
  • B affects the period. The period is 2π/|B| for sine and cosine and π/|B| for tangent and cotangent.
  • C causes a horizontal shift (phase shift).
  • D causes a vertical shift.
• f(x) = A cos(Bx + C) + D: Similar rules apply as for the sine function; period is 2π/|B|
• f(x) = A tan(Bx + C) + D: period is π/|B|.

Example 5: Finding the period of f(x) = 3sin(2x + π/4) – 1

Here, A = 3, B = 2, C = π/4, and D = -1. The period is only affected by B. So, the period is 2π/|2| = π.

Example 6: Finding the period of f(x) = -2cos(x/2 – π) + 5

Here, A = -2, B = 1/2, C = -π, and D = 5. The period is 2π/|1/2| = 4π.

Example 7: Finding the period of f(x) = 5tan(3x + π/3)

Here, A = 5, B = 3, C = π/3. The period is π/|3| = π/3.

3. Period of Sum/Difference of Periodic Functions

When you have a function formed by adding or subtracting two or more periodic functions, the period of the resulting function is the least common multiple (LCM) of the individual periods, if they have a common period.

The LCM rule

• If two periodic functions f(x) and g(x) have periods P1 and P2 respectively, then h(x) = f(x) + g(x) will be periodic with a period equal to LCM(P1, P2), provided that ratio of their periods is rational. If ratio of their periods is irrational then the resulting sum is not periodic.

Example 8: Finding the period of f(x) = sin(x) + cos(2x)

• The period of sin(x) is 2π.
• The period of cos(2x) is π.
• The LCM of 2π and π is 2π. Therefore, the period of f(x) is 2π.

Example 9: Finding the period of f(x) = sin(x) + sin(x/2)

• The period of sin(x) is 2π.
• The period of sin(x/2) is 4π.
• The LCM of 2π and 4π is 4π. Therefore, the period of f(x) is 4π.

Example 10: Finding the period of f(x) = sin(x) + cos(sqrt(2)x)

• The period of sin(x) is 2π.
• The period of cos(sqrt(2)x) is 2π/sqrt(2).
• Here the ratio of periods is (2π) / (2π/sqrt(2)) = sqrt(2) which is irrational and hence the function is not periodic

4. Using Graphs

Visualizing the graph of a function can often quickly reveal its period. This is especially helpful for complicated functions where analytical methods might be difficult to apply:

1. Graph the function: Use a graphing calculator or software (like Desmos or Wolfram Alpha) to plot the function.
2. Identify the repeating pattern: Look for a section of the graph that repeats itself.
3. Measure the distance: The length of the x-axis covered by one complete repetition represents the period.

This graphical approach works especially well when dealing with absolute value functions or piecewise-defined functions that might not have a standard formulaic approach.

5. Handling other non-Trigonometric functions

Most of the time you’ll come across periodic functions in the form of trigonometric functions and it’s variations. However, a few other non-trigonometric periodic functions are worth considering

Example 11: Finding the period of f(x) = x – floor(x)

• Here, floor(x) is a floor function which gives the greatest integer less or equal to x.
• This function essentially takes the decimal part of x.
• The function is periodic with a period of 1 because the decimal part of a number repeats every integer increment. For example, 1.2 – floor(1.2) = 0.2 and 2.2 – floor(2.2) = 0.2.
• By definition f(x+1) = (x+1) – floor(x+1) = x +1 – (floor(x)+1) = x – floor(x) = f(x). Hence period is 1

Important Notes and Tips

• Smallest Positive Period: When we say ‘the’ period of a function, we are always referring to the smallest positive value that satisfies the periodicity condition.
• Functions Without Period: Many functions, such as linear, polynomial (other than constant functions), and exponential functions, do not have a defined period.
• Rational vs. Irrational Period Ratios: When summing two periodic functions, if the ratio of their periods is a rational number (can be expressed as a fraction of integers), then the sum is also periodic. However, if the ratio is an irrational number, the sum is not periodic.
• Use Identities: Remember to utilize trigonometric identities or other algebraic manipulations when solving for P using the substitution method.
• Graphical verification: Always try to verify or get a hint by plotting a graph of the function and noting the interval after which it repeats
• Practice: The more examples you practice, the better you will become at recognizing patterns and finding periods.

Conclusion

Finding the period of a function is a valuable skill in mathematics and its applications. By understanding the basic definition of periodicity, knowing how transformations affect periods, and using methods like substitution, the LCM rule and graphical analysis, you can analyze a wide variety of periodic functions. Remember to always look for the smallest positive period and that not all functions are periodic. Practice with various types of examples to solidify your understanding and use graphing tools to help visualize the function’s periodic nature. With these tools and methods, you will be well-equipped to find the period of any periodic function you come across.