SAT Math December 8, 2023 · By Michael Rodriguez

Even and Odd Functions: The SAT Pattern You Must Know

Function symmetry is one of the most tested concepts on the Digital SAT Math section. Learn to identify even and odd functions algebraically and graphically—and solve these problems in under 30 seconds.

If you've seen SAT questions that mention f(-x) or ask about function symmetry, you're dealing with even and odd functions. This concept appears on nearly every Digital SAT Math section—and once you understand the pattern, these questions become some of the fastest points you can score.

Most students either don't know these definitions or confuse them. That's a mistake that costs points. In this guide, I'll break down exactly what even and odd functions are, how to test for them, and how to recognize them instantly on SAT graphs.

1. What Are Even and Odd Functions?

Even and odd functions are all about symmetry. The key is understanding what happens when you replace x with -x in the function.

Even Function Definition:

A function f(x) is even if f(-x) = f(x) for all x in the domain.

Symmetry: Even functions are symmetric about the y-axis.

Odd Function Definition:

A function f(x) is odd if f(-x) = -f(x) for all x in the domain.

Symmetry: Odd functions have 180° rotational symmetry about the origin.

Think of it this way:

  • Even functions: If you fold the graph along the y-axis, both halves match perfectly.
  • Odd functions: If you rotate the graph 180° around the origin, it looks identical.

2. The Algebraic Test

To determine whether a function is even, odd, or neither, use the algebraic test: replace x with -x and simplify.

Step-by-Step Process:

  1. Find f(-x) by substituting -x for every x in the function
  2. Simplify the expression completely
  3. Compare f(-x) to f(x):
    • If f(-x) = f(x), the function is even
    • If f(-x) = -f(x), the function is odd
    • If neither, the function is neither even nor odd

Example 1: Is f(x) = x2 + 4 even, odd, or neither?

f(x) = x2 + 4

f(-x) = (-x)2 + 4

f(-x) = x2 + 4

Since f(-x) = f(x), the function is even

Example 2: Is g(x) = x3 - 5x even, odd, or neither?

g(x) = x3 - 5x

g(-x) = (-x)3 - 5(-x)

g(-x) = -x3 + 5x

g(-x) = -(x3 - 5x) = -g(x)

Since g(-x) = -g(x), the function is odd

Example 3: Is h(x) = x2 + x even, odd, or neither?

h(x) = x2 + x

h(-x) = (-x)2 + (-x)

h(-x) = x2 - x

Since h(-x) ≠ h(x) and h(-x) ≠ -h(x), the function is neither

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3. Recognizing Symmetry on Graphs

On the Digital SAT, you'll often be shown a graph and asked whether it represents an even function, odd function, or neither. The visual test is much faster than algebra.

Visual Test for Even Functions:

Imagine folding the graph along the y-axis. If the left and right halves match perfectly, it's even.

Examples: y = x2, y = x4, y = cos(x), y = |x|

Visual Test for Odd Functions:

Imagine rotating the graph 180° around the origin. If it looks the same after rotation, it's odd.

Examples: y = x, y = x3, y = x5, y = sin(x)

Quick Recognition Tips:

  • Even functions: Both sides of the y-axis look like mirror images
  • Odd functions: If a point (a, b) is on the graph, then (-a, -b) must also be on the graph
  • Neither: If you don't see clear symmetry in either direction, it's neither

SAT Tip: On graph questions, check a few key points. If (2, 4) is on the graph:

  • For even: look for (-2, 4)
  • For odd: look for (-2, -4)

4. Common Examples and Patterns

Memorizing these common patterns will save you precious time on test day. Here are the functions you're most likely to see:

Even Functions:

  • f(x) = x2
  • f(x) = x4
  • f(x) = x6
  • f(x) = |x|
  • f(x) = cos(x)
  • f(x) = x2 + 3

Pattern: All even-powered terms only

Odd Functions:

  • f(x) = x
  • f(x) = x3
  • f(x) = x5
  • f(x) = sin(x)
  • f(x) = x3 - 5x
  • f(x) = 2x7 + x3

Pattern: All odd-powered terms only

Neither Even Nor Odd:

  • f(x) = x2 + x (mix of even and odd powers)
  • f(x) = x3 + 2 (odd power plus constant)
  • f(x) = 2x (exponential function)

Pattern: Mix of even/odd powers, or exponentials/logs

Important Rule: A polynomial is even only if all its terms have even exponents. A polynomial is odd only if all its terms have odd exponents. If there's any mix, it's neither.

Key Takeaways

  • Even functions: f(-x) = f(x), symmetric about the y-axis
  • Odd functions: f(-x) = -f(x), symmetric about the origin
  • Quick graphical check: Mirror image across y-axis = even; 180° rotation = odd
  • Polynomial pattern: All even powers = even, all odd powers = odd, mixed = neither

Master these patterns and you'll breeze through function symmetry questions on the SAT. Remember: the Digital SAT rewards pattern recognition, not lengthy calculations.

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