Define odd function and even function.

In mathematics, functions can be classified as either odd or even based on their symmetry properties. Understanding these types helps in analyzing their behavior.

Even Functions

An even function is defined by the property that its output remains the same when the input is negated. Mathematically, this is expressed as:

f(-x) = f(x)

This means that the graph of an even function is symmetric with respect to the y-axis. Common examples include:

• f(x) = x²
• f(x) = cos(x)

Odd Functions

Conversely, an odd function has the property that its output is the negative of the input when negated. This can be written as:

f(-x) = -f(x)

The graph of an odd function exhibits symmetry about the origin. Examples of odd functions include:

• f(x) = x³
• f(x) = sin(x)

Summary

In summary, even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. Recognizing these properties can be very useful in various areas of mathematics, including calculus and graphing.