Classifying Polynomial Functions as Even, Odd, or Neither
Classifying polynomial functions as even, odd, or neither is a Grade 11 algebra skill in Big Ideas Math based on symmetry analysis. An even function satisfies f(−x) = f(x) for all x, producing y-axis symmetry (e.g., f(x) = x⁴ − 2x²). An odd function satisfies f(−x) = −f(x), producing origin symmetry (e.g., f(x) = x³ − x). A function is neither if it fails both tests (e.g., f(x) = x² + x). To classify: substitute −x for x, simplify, and compare to f(x) and −f(x). All exponents must be even for an even function; all odd for an odd function; a mix gives neither.
Key Concepts
A function $f(x)$ is even if $f( x) = f(x)$ for all $x$ in the domain (symmetric about the y axis).
A function $f(x)$ is odd if $f( x) = f(x)$ for all $x$ in the domain (symmetric about the origin).
Common Questions
What is the test for an even function?
A function is even if f(−x) = f(x) for all x in the domain. Graphically, even functions have y-axis symmetry.
What is the test for an odd function?
A function is odd if f(−x) = −f(x) for all x in the domain. Graphically, odd functions have origin symmetry (rotational symmetry of 180°).
Is f(x) = x⁴ − 2x² even, odd, or neither?
f(−x) = (−x)⁴ − 2(−x)² = x⁴ − 2x² = f(x). Since f(−x) = f(x), the function is even.
Is f(x) = x³ − x even, odd, or neither?
f(−x) = (−x)³ − (−x) = −x³ + x = −(x³ − x) = −f(x). Since f(−x) = −f(x), the function is odd.
Is f(x) = x² + x even, odd, or neither?
f(−x) = x² − x. This equals neither f(x) = x²+x nor −f(x) = −x²−x. The function is neither even nor odd.
What shortcut works for polynomial functions?
A polynomial is even if all its terms have even exponents (plus constant). It is odd if all terms have odd exponents. A mix of even and odd exponents makes it neither.