What is an odd function?

An odd function is a type of function that exhibits symmetry about the origin. Here's a breakdown:

• Definition: A function f(x) is considered odd if it satisfies the condition: f(-x) = -f(x) for all x in its domain. This is in contrast to an even function, where f(-x) = f(x).

• Symmetry: Odd functions have rotational symmetry of 180 degrees about the origin. This means if you rotate the graph of the function 180 degrees around the origin, it will look the same.

• Graphical Representation: The graph of an odd function will always pass through the origin (0,0), unless it has a discontinuity at x=0. This is because f(0) = -f(0), which implies f(0) = 0.

• Examples: Common examples of odd functions include:

  • f(x) = x
  • f(x) = x³
  • f(x) = sin(x)
  • f(x) = tan(x)

• Operations:

  • The sum or difference of two odd functions is also an odd function.
  • The product or quotient of two odd functions is an even function.
  • The product or quotient of an odd function and an even function is an odd function.

• Integral Properties: The definite integral of an odd function over a symmetric interval [-a, a] is always zero. That is, ∫[-a, a] f(x) dx = 0. This is a useful property in calculus.