Symmetry Analyzer

Math Algebra • Functions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

Enter \(f(x)\) and press Calculate. The tool will classify the function as:

Even if \(f(-x)=f(x)\)
Odd if \(f(-x)=-f(x)\)
Neither otherwise

Blue: \(f(x)\). Orange (dashed): \(f(-x)\). Pink (dashed): \(-f(x)\). Drag to pan. Mouse wheel to zoom. Double-click to reset.

Number line: probe point \(x_0\) and its symmetric partner \(-x_0\).

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6. Symmetry Analyzer

Symmetry is a global “shape property” of a function. Even/odd symmetry can often simplify graphs, integrals, Fourier analysis, and reasoning about behavior.

1) Even and odd functions

\[ \textbf{Even: }\; f(-x)=f(x)\quad\text{(y-axis symmetry)} \] \[ \textbf{Odd: }\; f(-x)=-f(x)\quad\text{(origin symmetry)} \]

If \(f\) is even, the graph reflects across the \(y\)-axis.
If \(f\) is odd, the graph is symmetric by a 180° rotation about the origin.
The zero function \(f(x)=0\) is both even and odd.

2) How the calculator tests symmetry

A strict algebraic proof may require symbolic manipulation. This calculator uses a robust numerical check on a chosen window:

\[ E(x)=|f(-x)-f(x)|,\qquad O(x)=|f(-x)+f(x)|. \]

If the maximum error stays below a small tolerance (scaled to typical function size), it reports “Even” or “Odd” on that window. This is a strong practical test, but it is still a numerical test (not a formal proof).

3) Periodicity (optional check)

A function is periodic with period \(T>0\) if:

\[ f(x+T)=f(x)\quad \text{for all } x \text{ in its domain.} \]

The calculator can test a user-provided period \(T\), or try common candidates like \(2\pi\), \(\pi\), \(\pi/2\) (useful for trig-style functions).

If you see “Not periodic”, it might still be periodic outside the chosen window, or the window may not allow enough overlap for \(x\) and \(x+T\).

4) Visual overlay idea

Overlay \(f(x)\) with \(f(-x)\): if they match, the function is even.
Overlay \(f(x)\) with \(-f(x)\) compared to \(f(-x)\): if \(f(-x)\) matches \(-f(x)\), the function is odd.

5) Quick examples

\[ x^2,\ \cos(x),\ |x|\ \text{ are even.} \] \[ x^3,\ \sin(x)\ \text{ are odd.} \] \[ e^x\ \text{is neither even nor odd.} \]

Frequently Asked Questions

What does it mean for a function to be even or odd?

A function is even when f(-x)=f(x), which corresponds to symmetry about the y-axis. A function is odd when f(-x)=-f(x), which corresponds to symmetry about the origin.

How does the symmetry analyzer decide if f(x) is even or odd?

It evaluates f(x) and f(-x) at many points in the selected window and measures the size of f(-x)-f(x) and f(-x)+f(x). If the maximum error stays below a small tolerance, it reports even or odd for that window.

Why can a function be neither even nor odd?

If f(-x) does not match f(x) and also does not match -f(x) across the tested window, then the graph has neither y-axis symmetry nor origin symmetry. Many functions like exp(x) fall into this category.

What does the periodicity option check?

It tests whether f(x+T)=f(x) holds numerically on the window for a chosen period T. A function can look non-periodic on a limited window even if it is periodic on a broader domain overlap.

Can a function be both even and odd?

Only the zero function f(x)=0 satisfies both f(-x)=f(x) and f(-x)=-f(x) for all x. Otherwise, a nonzero function cannot be both even and odd.