Determining If Functions Are Even, Odd, or Neither

The Organic Chemistry Tutor

This comprehensive mathematics tutorial provides a clear, step-by-step guide on how to classify mathematical functions as even, odd, or neither. The video breaks down the concept into two primary methods: algebraic analysis and graphical interpretation. It begins by defining the formal algebraic conditions for even functions ($f(-x) = f(x)$) and odd functions ($f(-x) = -f(x)$), while also introducing a practical "shortcut" involving the exponents of the variables. The narrator walks through multiple algebraic examples, demonstrating how to substitute negative variables and factor equations to prove a function's classification. The second half of the video shifts to visual learning, explaining how symmetry on a coordinate plane identifies function types. It demonstrates that even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. The video illustrates these concepts with hand-drawn graphs of parabolas, cubic functions, lines, and complex polynomials. It visually emphasizes how translations (shifts) affect symmetry and changes a function's classification from even/odd to "neither." For educators and students, this resource is invaluable for Algebra II and Pre-Calculus curricula. It addresses common stumbling blocks, such as how to handle constants (treating them as $x^0$) and linear terms (treating $x$ as $x^1$). The video concludes with a critical conceptual test using a circle, challenging students to recall the definition of a function itself (the Vertical Line Test) before attempting to classify it. This encourages critical thinking beyond rote memorization of rules.