https://www.pw.live/study-v2/notes?pdf=https://static.pw.live/5eb393ee95fab7468a79d189/ADM and Function - Lecture 9: Even, Odd, and Periodic Functions

Overview of Relation and Function - Lecture 9

• Course: Power Batch - JEE Mathematics

• Instructor: Ravindra Bahadur Sir

• Main Topics: Even and Odd Functions, Fundamental Properties of Symmetry, and Introduction to Periodic Functions.

• Goal: To establish the definitive properties of functions based on their symmetry (parity) and their repetition (periodicity).

Even and Odd Functions: Definitions

• Even Function:

  • A function $y = f(x)$ is classified as an even function if substituting $-x$ for $x$ leaves the function unchanged.

  • Condition: $f(-x) = f(x)$.

  • Graphical Property: The graph of an even function is symmetrical about the $y$-axis.

• Odd Function:

  • A function $y = f(x)$ is classified as an odd function if substituting $-x$ for $x$ results in the negative of the original function.

  • Condition: $f(-x) = -f(x)$.

  • Graphical Property: The graph of an odd function is symmetrical about the origin (rotational symmetry of $180^\text{o}$).

• NENO (Neither Even Nor Odd):

  • If a function does not satisfy either of the above conditions ($f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$), it is categorized as Neither Even Nor Odd (NENO).

Examples of Parity Verification

• Trigonometric Functions:

  • $f(x) = \tan(x)$: $f(-x) = \tan(-x) = -\tan(x) = -f(x)$. Conclusion: Odd.

  • $f(x) = \text{cos}(x)$: $f(-x) = \text{cos}(-x) = \text{cos}(x) = f(x)$. Conclusion: Even.

  • $f(x) = \text{sin}(x)$: $f(-x) = \text{sin}(-x) = -\text{sin}(x) = -f(x)$. Conclusion: Odd.

• Algebraic Combinations:

  • $f(x) = x^3$: $f(-x) = (-x)^3 = -x^3 = -f(x)$. Conclusion: Odd.

  • $f(x) = x^3 \text{sin}(x)$:

    • $f(-x) = (-x)^3 \text{sin}(-x) = (-x^3)(-\text{sin}(x)) = x^3 \text{sin}(x) = f(x)$.

    • Conclusion: Even (Product of two odd functions is even).

  • $f(x) = x^2$: Conclusion: Even.

  • $f(x) = x^3 + 2x - 1$:

    • $f(-x) = (-x)^3 + 2(-x) - 1 = -x^3 - 2x - 1 = -(x^3 + 2x + 1)$.

    • Since $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, it is NENO.

• Transcendental Functions:

  • $f(x) = \frac{e^x - 1}{e^x + 1}$:

    • $f(-x) = \frac{e^{-x} - 1}{e^{-x} + 1} = \frac{\frac{1}{e^x} - 1}{\frac{1}{e^x} + 1} = \frac{1 - e^x}{1 + e^x} = -\frac{e^x - 1}{e^x + 1} = -f(x)$.

    • Conclusion: Odd.

  • $f(x) = x \frac{e^x - 1}{e^x + 1}$: Product of Odd ($x$) and Odd ($\frac{e^x - 1}{e^x + 1}$). Conclusion: Even.

The Signum Function and Parity

• Definition of $y = \text{sgn}(x)$:     $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$

• Note: The Signum function is always an Odd Function ($\text{sgn}(-x) = -\text{sgn}(x)$).

• Domain: $x \times \text{R}$

• Range: $\{-1, 0, 1\}$

• Complex Transformation Example:

  • $f(x) = \text{sgn}(x^2 + 2x + 3)$.

  • Analyze the inner expression: $x^2 + 2x + 3 = (x+1)^2 + 2$. This expression is always strictly greater than zero for all real $x$.

  • Since the input to the signum function is always positive, $\text{sgn}(\text{positive}) = 1$.

  • $f(x) = 1$ for all $x$.

  • Range: $\{1\}$.

  • Conclusion: Even (Constant function).

Properties of Even and Odd Functions

Let $E$ represent an Even function and $O$ represent an Odd function:

• Addition and Subtraction:

  • $E \times E = E$

  • $O \times O = O$

  • $O \times E = \text{NENO}$

• Multiplication and Division:

  • $E \times E = E$

  • $O \times O = E$ (The product of two odd functions is always even)

  • $O \times E = O$ (The product of an odd and even function is always odd)

• Special Cases:

  • If a function is both Even and Odd simultaneously, then $f(x) = -f(x) \rightarrow 2f(x) = 0 \rightarrow f(x) = 0$. The zero function is the only function that is both even and odd.

Deep Dive Questions and Problem Solving

• Question 1: Parity of $f(x) = \text{ln}(x + \text{sqrt}(x^2 + 1))$

  • $f(-x) = \text{ln}(-x + \text{sqrt}((-x)^2 + 1)) = \text{ln}(\text{sqrt}(x^2 + 1) - x)$.

  • Multiply and divide by the conjugate: $\frac{(\text{sqrt}(x^2 + 1) - x)(\text{sqrt}(x^2 + 1) + x)}{\text{sqrt}(x^2 + 1) + x} = \frac{x^2 + 1 - x^2}{\text{sqrt}(x^2 + 1) + x} = \frac{1}{\text{sqrt}(x^2 + 1) + x}$.

  • $f(-x) = \text{ln}(\frac{1}{x + \text{sqrt}(x^2 + 1)}) = \text{ln}(x + \text{sqrt}(x^2 + 1))^{-1} = -\text{ln}(x + \text{sqrt}(x^2 + 1)) = -f(x)$.

  • Answer: An odd function.

• Question 2: Parameter finding for $f: [-20, 20] \rightarrow \text{R}$

  • Given: $f(x) = \frac{x^2}{[\frac{x^2}{a}] + 1}$ is an even function. (Note: $[.]$ is G.I.F.).

  • For the function to be even over the domain $[-20, 20]$, the term $[\frac{x^2}{a}]$ must remain constant or follow an even pattern that doesn't violate functional definitions.

  • Typically, if $[\frac{x^2}{a}] = 0$, the function reduces to $f(x) = x^2$, which is even.

  • Max value of $x^2$ on $[-20, 20]$ is $x^2 = 400$.

  • For $[\frac{x^2}{a}] = 0$, we need 0 \times \frac{x^2}{a} < 1. Substituting max value: \frac{400}{a} < 1 \rightarrow a > 400.

  • Answer: $(400, \text{infinity})$.

• Question 3: Simultaneously Even and Odd Function

  • If $f(x)$ is odd and even, $f(3) - f(2)$?

  • As established, if a function is both, $f(x) = 0$ for all $x$.

  • $f(3) = 0$ and $f(2) = 0$.

  • Result: $0 - 0 = 0$.

  • Answer: 0.

Introduction to Periodic Functions

• Definition: A function $f: X \rightarrow Y$ is periodic if there exists a positive real number $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain.

• Fundamental Period: The smallest positive value of $T$ is called the fundamental period (or principal period).

• Graphical repetition: The graph repeats its shape identically over every interval of length $T$.

Fundamental Periods of Standard Functions

Periodic Function Exercises

1. Period of $f(x) = e^{\text{sin}(x)}$:

   • The outer function $e^u$ is not periodic, but the inner function $\text{sin}(x)$ is periodic with $2\text{pi}$.

   • $f(x + 2\text{pi}) = e^{\text{sin}(x + 2\text{pi})} = e^{\text{sin}(x)} = f(x)$.

   • Answer: $2\text{pi}$.

2. Period of $f(x) = \text{tan}([x])$:

   • Here $[x]$ is the greatest integer function. Since the input to $\text{tan}$ changes in discrete steps and does not repeat in a fixed real interval pattern $f(x+T)=f(x)$, it is not periodic.

3. Period of $f(x) = [6x + 7] - 6x$:

   • Simplify: $[6x + 7] - 6x = [6x] + 7 - 6x = 7 - (6x - [6x]) = 7 - \{6x\}$.

   • The period of $\{x\}$ is $1$. Therefore, the period of $\{kx\}$ is $\frac{1}{|k|}$.

   • Period of $\{6x\} = \frac{1}{6}$.

   • Answer: $\frac{1}{6}$.

4. Complex Summations ($x$ terms):

   • $f(x) = [x] + [2x] + [3x] + \text{...} + [nx] - \frac{n(n+1)}{2}x$.

   • Recognize that $\frac{n(n+1)}{2}x = x + 2x + 3x + \text{...} + nx$.

   • The function becomes: $([x] - x) + ([2x] - 2x) + \text{...} + ([nx] - nx) = -(\{x\} + \{2x\} + \text{...} + \{nx\})$.

   • Individual periods are $1, \frac{1}{2}, \frac{1}{3}, \text{...}, \frac{1}{n}$.

   • The period is the L.C.M. of the individual periods: $\text{LCM}(1, \frac{1}{2}, \frac{1}{3}, \text{...}, \frac{1}{n}) = 1$.

   • Answer: 1.

Questions & Homework (H.W.)

• Identify parity for: $f(x) = \{g(x) - g(-x)\}^3$.

• Signum Logic Homework: Given $f(x) = e^{\{|x| / \text{sgn}(x)\}}$ and $g(x) = e^{[|x| \text{sgn}(x)]}$. Determine the parity of $h(x) = \text{log}(f(x)) + \text{log}(g(x))$.

• Greatest Integer Function Parity: Check parity for $f(x) = \frac{x(\text{sin}(x) + \text{tan}(x))}{[\frac{x + \text{pi}}{\text{pi}}] - \frac{1}{2}}$.

• Find Value of $n$: If $f(x) = \text{tan}(\text{sqrt}([n])x)$ has a period of $\text{pi}/3$, find the integer value of $n$.

• Coefficient Problem: If $f(x) = \text{lambda}|\text{sin}(x)| + \text{lambda}^2|\text{cos}(x)| + g(\text{lambda})$ has a period of $\text{pi}/2$, solve for $\text{lambda}$.