Even and Odd Functions

Symmetry in Functions

Some functions have special symmetry properties.

• Even functions — symmetric about the y-axis
• Odd functions — symmetric about the origin

Even Functions

A function is even if:

$f(-x) = f(x)$

Plugging in $-x$ gives the same result as plugging in $x$.

Examples of even functions:

• $f(x) = x^2$
• $f(x) = x^4$
• $f(x) = |x|$
• $f(x) = \cos(x)$

Visual test: If you fold the graph along the y-axis, the two halves match.

Odd Functions

A function is odd if:

$f(-x) = -f(x)$

Plugging in $-x$ gives the negative of $f(x)$.

Examples of odd functions:

• $f(x) = x$
• $f(x) = x^3$
• $f(x) = x^5$
• $f(x) = \sin(x)$

Visual test: If you rotate the graph 180° about the origin, it looks the same.

Testing Algebraically

To determine if a function is even, odd, or neither:

1. Compute $f(-x)$
2. Compare to $f(x)$ and $-f(x)$

Example: Is $f(x) = x^4 - 2x^2$ even, odd, or neither?

Step 1: Find $f(-x)$

Step 2: Compare

• $f(-x) = x^4 - 2x^2$
• $f(x) = x^4 - 2x^2$

They’re equal! → Even

Example: Is $f(x) = x^3 - x$ even, odd, or neither?

Step 1: Find $f(-x)$

Step 2: Compare to $-f(x)$

$-f(x) = -(x^3 - x) = -x^3 + x$

$f(-x) = -f(x)$ → Odd

Example: Is $f(x) = x^2 + x$ even, odd, or neither?

Step 1: Find $f(-x)$

Step 2: Compare

• $f(-x) = x^2 - x$
• $f(x) = x^2 + x$
• $-f(x) = -x^2 - x$

$f(-x)$ equals neither $f(x)$ nor $-f(x)$ → Neither

Quick Patterns

For polynomials:

• All even powers → even function ($x^4 + x^2 + 1$)
• All odd powers → odd function ($x^5 - x^3 + x$)
• Mixed powers → usually neither ($x^3 + x^2$)

Properties

Multiplication:

• Even × Even = Even
• Odd × Odd = Even
• Even × Odd = Odd

Addition:

• Sum of even functions → even
• Sum of odd functions → odd
• Sum of even + odd → usually neither