Title: How to Determine Whether a Function is Even, Odd, or Neither – With Graphs and Algebra

This post breaks down the process of identifying whether a function is even, odd, or neither—using both algebraic rules and graph symmetry. This concept shows up across algebra, precalculus, and calculus exams.

The Problem:
Determine whether each function is even, odd, or neither:

(a) f(x) = 2x⁵ – 3x² + 2
(b) f(x) = x³ – x⁷
(c) f(x) = (1 – x²) / (1 + x²)
(d) f(x) = 1 / (x + 2)

Step-by-Step Guide:

1. Use the Algebraic Test:

• Even: f(–x) = f(x)

• Odd: f(–x) = –f(x)

• Neither: f(–x) ≠ ±f(x)

2. Apply the Formula to Each Function:

• (a) f(x) = 2x⁵ – 3x² + 2 → f(–x) ≠ f(x), ≠ –f(x) → Neither

• (b) f(x) = x³ – x⁷ → f(–x) = –f(x) → Odd

• (c) f(x) = (1 – x²) / (1 + x²) → f(–x) = f(x) → Even

• (d) f(x) = 1 / (x + 2) → f(–x) ≠ f(x), ≠ –f(x) → Neither

3. Verify with Graph Symmetry:

• Even functions are symmetric about the y-axis.

• Odd functions are symmetric about the origin.

• Neither functions show no obvious symmetry.

The visuals on the second page compare all four graphs and highlight their symmetry properties, giving you an intuitive way to verify the result alongside your algebra.

This material is from Chapter 3: Functions of the Ultimate Crash Course for STEM Majors. It shows how to earn full credit on tests by identifying properties of functions both visually and algebraically.

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