Overview
This lecture explains how to determine if a function is even, odd, or neither using algebraic methods and graph symmetry, with examples and key definitions.
Definitions and Rules
- A function is even if f(-x) = f(x); its graph is symmetric about the y-axis.
- A function is odd if f(-x) = -f(x); its graph is symmetric about the origin.
- If a function is neither even nor odd, f(-x) ≠ f(x) and f(-x) ≠ -f(x).
- For polynomials: all even exponents suggest an even function; all odd exponents suggest an odd function; a mix means neither.
Algebraic Tests for Even, Odd, or Neither
- To test evenness, replace x with -x; if the function remains unchanged, it is even.
- To test oddness, replace x with -x; if every term changes sign (factor out a -1), and it matches -f(x), the function is odd.
- If neither test succeeds, the function is neither even nor odd.
Worked Examples
- f(x) = x⁴ + 3x²: All exponents even, so function is even (f(-x) = f(x)).
- f(x) = x⁵ + 2x³: All exponents odd, so function is odd (f(-x) = -f(x)).
- f(x) = x² + 6: Constant terms like 6 are even (as 6x⁰), so the function is even.
- f(x) = x³ - 8x: All exponents odd, so function is odd.
- f(x) = x³ - 5x² + 2: Mix of even and odd exponents, so function is neither even nor odd.
Graphical Interpretation
- Even functions: Graph symmetric about y-axis (e.g., y = x², y = 3).
- Odd functions: Graph symmetric about the origin (e.g., y = x³, y = x).
- Neither: No symmetry about y-axis or origin.
- For circles: Although symmetric, a circle is not a function (does not pass the vertical line test).
Key Terms & Definitions
- Even function — Satisfies f(-x) = f(x), symmetric about y-axis.
- Odd function — Satisfies f(-x) = -f(x), symmetric about the origin.
- Neither — Fails both even and odd conditions.
- Vertical line test — Used to determine if a graph represents a function.
Action Items / Next Steps
- Practice determining if functions are even, odd, or neither using algebraic substitutions.
- Review graph symmetry for quick visual identification.
- Complete assigned exercises or homework related to this topic if provided.