Even function

Definition

An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.

5 Must Know Facts For Your Next Test

1. The graph of an even function is symmetric with respect to the y-axis.
2. If $f(x)$ is a polynomial, it is even if all powers of $x$ are even (e.g., $f(x) = x^2 + 4$).
3. Common examples of even functions include $f(x) = x^2$ and $f(x) = \cos(x)$.
4. The sum or difference of two even functions is also an even function.
5. A function can be neither even nor odd if it doesn't satisfy the conditions for either.

Review Questions

• What condition must an even function satisfy?
• How does the graph of an even function relate to the y-axis?
• Give an example of a basic polynomial that represents an even function.

Related terms

Odd Function: A function $g(x)$ where $g(-x) = -g(x)$ for all $x$ in its domain, showing symmetry about the origin.

Symmetry: In mathematics, symmetry refers to a situation where one part of a shape or object exactly mirrors another part.

$y$-axis: $The vertical axis in a coordinate plane, used as a reference line for measuring horizontal distances.$