Even and Odd Functions
Why Does This Matter?
Understanding even and odd functions helps us recognize symmetry in mathematics. This concept simplifies graphing, integration in calculus, and appears throughout physics and engineering. Once you learn to spot these patterns, you'll see them everywhere!
📐 Even Functions
Definition
A function $f(x)$ is even if for every $x$ in its domain:
$$f(-x) = f(x)$$
Graphical Meaning: An even function is symmetric about the y-axis. If you fold the graph along the y-axis, the two halves match perfectly.
Common Examples:
- $f(x) = x^2$
- $f(x) = x^4$
- $f(x) = |x|$
- $f(x) = \cos(x)$
- Any polynomial with only even powers of $x$ (including constants)
Memory Tip: "Even" functions have "even" exponents! Think of $x^2, x^4, x^6, \ldots$
🔄 Odd Functions
Definition
A function $f(x)$ is odd if for every $x$ in its domain:
$$f(-x) = -f(x)$$
Graphical Meaning: An odd function has rotational symmetry about the origin (180° rotation). If you rotate the graph 180° around the origin, it looks the same.
Common Examples:
- $f(x) = x$
- $f(x) = x^3$
- $f(x) = x^5$
- $f(x) = \sin(x)$
- Any polynomial with only odd powers of $x$
Memory Tip: "Odd" functions have "odd" exponents! Think of $x, x^3, x^5, \ldots$