Odd and Even Functions

Definition and Key Characteristics

• Even Functions:
  • A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain.
  • Graphical Symmetry: The graph is symmetric about the y-axis.

• Odd Functions:
  • A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.
  • Graphical Symmetry: The graph is symmetric about the origin.

Periodic Functions

• A function \( f(x) \) is periodic if \( f(x + T) = f(x) \) for some \( T > 0 \), where \( T \) is the period.
• Even periodic functions: Examples include \( f(x) = \cos(x) \), which satisfies \( \cos(-x) = \cos(x) \).

• Odd periodic functions: Examples include \( f(x) = \sin(x) \), which satisfies \( \sin(-x) = -\sin(x) \).

Finding the Inverse Function \( f^{-1}(x) \)

• Definition: The inverse function reverses the effect of the original function. If \( f(a) = b \), then \( f^{-1}(b) = a \).
• To find \( f^{-1}(x) \):
  • Swap \( x \) and \( y \) in the equation \( y = f(x) \).
  • Solve for \( y \) in terms of \( x \).
  • Ensure that the function is one-to-one (injective) on the restricted domain.
• Example: For \( f(x) = x^2 \), the inverse is \( f^{-1}(x) = \sqrt{x} \), but the domain must be restricted to \( x \geq 0 \).

Self-Inverse Functions

• A function is self-inverse if \( f(f(x)) = x \) for all \( x \) in its domain.
• Such functions are their own inverses, meaning \( f^{-1}(x) = f(x) \).
• Example: The reciprocal function \( f(x) = \frac{1}{x} \), where \( x \neq 0 \), is self-inverse.

Properties and Applications

• Even and odd functions help identify symmetry and simplify integrals in calculus.
• Periodic functions are widely used in modeling natural phenomena like sound waves and tides.
• Inverse functions are essential in solving equations and understanding function transformations.
• Self-inverse functions are foundational in fields like cryptography and geometry.