Even Functions

Definition

A function $f$ is even if $f(-x) = f(x)$ for every $x$ in its domain. The graph of an even function is symmetric about the $y$-axis.

• How to read: “The function evaluated at negative x equals f of x.”
• Meaning: Replacing x with its opposite leaves the output unchanged—mirror symmetry across the y-axis.

Why It Matters

Identifying even symmetry in functions allows for massive simplifications in calculus, such as doubling the integral of a function over a symmetric interval $[ -a, a ]$. It provides a cognitive shortcut for graphing and analysis, revealing the underlying balance of a mathematical or physical system.

Core Concepts

• Symmetry Test: To test if a function is even, substitute $-x$ for $x$ and simplify. If the result is the original function, it is even.
• Polynomials: Polynomials with only even powers of $x$ (including constants, which are $x^0$) are even functions.
  • Example: $f(x) = x^2$ is even because $(-x)^2 = x^2$.
    • How to read: “The quantity negative x squared equals x squared.”
    • Meaning: Squaring erases the sign—classic even polynomial.
• Trigonometric Functions: The cosine function is a primary example of an even function.
  • Example: $f(x) = \cos x$ is an even function.
    • How to read: “The cosine of negative x equals the cosine of x.”
    • Meaning: Cosine is y-axis symmetric on the unit circle.

Connected Concepts

• Function Definition, Function Notation
• Function Domain
• Function Range
• Graphs of Functions: Properties
• Odd Functions