Understanding Odd Functions | Symmetry Around the Origin and Key Properties

Odd function

An odd function is a mathematical function that exhibits symmetry around the origin, meaning that the function’s values are mirrored across the y-axis

An odd function is a mathematical function that exhibits symmetry around the origin, meaning that the function’s values are mirrored across the y-axis. In other words, if you were to fold the graph of an odd function along the y-axis, the two halves would coincide.

Formally, a function f(x) is considered odd if it satisfies the property: f(-x) = -f(x) for all x in the domain of the function.

This definition implies that the output (or the value of y) for the opposite input values (x and -x) have opposite signs. For example, if f(2) = 4, then f(-2) = -4. This behavior creates a symmetrical pattern on the graph.

The graph of an odd function typically exhibits symmetry across the origin. If a point (x, y) lies on the graph, then its reflection (-x, -y) also lies on the graph. Common examples of odd functions include f(x) = x, f(x) = x^3, and f(x) = sin(x).

Some key properties of odd functions include:

1. The product of two odd functions is an even function. (f(x) * g(x) = g(-x) * f(-x) = f(-x) * g(-x))

2. The composition of an odd function with an even function is an odd function. (f(g(x)) = -f(x))

3. The integral of an odd function over a symmetric interval around the origin is zero. (∫[-a, a] f(x) dx = 0)

Knowing whether a function is odd or even helps in simplifying calculations and understanding the symmetry and behavior of functions and their graphs.

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