Overview
This lecture explains how to determine if a function is even, odd, or neither, using algebraic methods and graph symmetry, with step-by-step examples.
Algebraic Tests for Even, Odd, or Neither
- A function is even if f(-x) = f(x) for all x (all exponents of x are even).
- A function is odd if f(-x) = -f(x) for all x (all exponents of x are odd).
- If f(-x) ≠ f(x) and f(-x) ≠ -f(x), the function is neither even nor odd.
- Coefficients (like 2 in 2x) do not affect whether a function is even or odd; only exponents matter.
- A constant (e.g., f(x) = 6) is considered even (since x⁰ = 1, and 0 is even).
Example Problems
- f(x) = x⁴ + 3x²: All exponents even, so the function is even.
- f(x) = x⁵ + 2x³: All exponents odd, so the function is odd.
- f(x) = x² + 6: x² is even and 6 is a constant (even), so the function is even.
- f(x) = x³ - 8x: Both terms have odd exponents, so the function is odd.
- f(x) = x³ - 5x² + 2: Mix of even and odd exponents, so the function is neither.
Proving Algebraically
- Substitute x with -x and simplify to check if the resulting function matches f(x), -f(x), or neither.
- If after substitution, only some terms change sign, the function is neither.
Graphing and Symmetry
- Even functions are symmetric about the y-axis (the left and right sides look the same).
- Odd functions are symmetric about the origin (rotation of 180° gives the same graph).
- Neither functions lack both y-axis and origin symmetry.
- Constant functions graph as horizontal lines and are even.
- The graph of y = x² is a symmetric "U" (even).
- The graph of y = x³ or y = x is symmetric about the origin (odd).
- Non-functions (like a circle) may appear symmetric, but they fail the vertical line test, so they are not considered even or odd functions.
Key Terms & Definitions
- Even Function — A function satisfying f(-x) = f(x) for all x in its domain.
- Odd Function — A function satisfying f(-x) = -f(x) for all x in its domain.
- Neither — A function that is not even nor odd.
- Symmetry about y-axis — Both sides of the graph match across the y-axis (even function property).
- Symmetry about origin — Graph matches after 180° rotation around the origin (odd function property).
- Vertical Line Test — A graph represents a function only if no vertical line crosses it more than once.
Action Items / Next Steps
- Practice by classifying functions as even, odd, or neither using substitution and graph symmetry.
- Review homework or textbook problems on function symmetry.
- Prepare examples to discuss in the next class.