Properties of Even and Odd Functions
Quick Recap: Definitions
A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain. (Symmetric about the $y$-axis)
A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain. (Symmetric about the origin)
Visual Reference: Even vs. Odd Functions
Interactive visualization — available in the full lesson
Common Examples of Even and Odd Functions
Even Functions
- $y = b$ (any constant function)
- $y = x^2$, $y = x^4$, $y = x^{2n}$ (even powers)
- $y = |x|$ (absolute value)
- $y = \cos(x)$
Key idea: $f(-x) = f(x)$ — the output doesn't change when you negate the input.
Odd Functions
- $y = kx$ (direct proportion / linear through origin)
- $y = k/x$ (inverse proportion)
- $y = x^3$, $y = x^5$, $y = x^{2n-1}$ (odd powers)
- $y = \text{sgn}(x)$ (sign function)
- $y = \sin(x)$
Key idea: $f(-x) = -f(x)$ — negating the input negates the output.