1. Let's start by stating the problem: We want to understand the difference between odd and even functions. 2. An **even function** satisfies the property: $$f(-x) = f(x)$$ for all $x$ in the domain. This means the function is symmetric about the y-axis. 3. An **odd function** satisfies the property: $$f(-x) = -f(x)$$ for all $x$ in the domain. This means the function is symmetric about the origin. 4. Important rules: - Even functions have mirror symmetry on the y-axis. - Odd functions have rotational symmetry of 180 degrees about the origin. - A function can be neither even nor odd. - The zero function $f(x) = 0$ is both even and odd. 5. Examples: - Even: $f(x) = x^2$ because $(-x)^2 = x^2$ - Odd: $f(x) = x^3$ because $(-x)^3 = -x^3$ 6. To check if a function is even or odd, substitute $-x$ into the function and compare with $f(x)$ and $-f(x)$. This understanding helps in graphing and analyzing functions in algebra and calculus.