Even or Odd Decomposition

Math Algebra • Functions

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

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8. Even Or Odd Decomposition

Math Algebra • Functions

Any function \(f(x)\) can be split into an even part and an odd part: \[ f(x) = e(x) + o(x), \] where \(e(x)\) is even and \(o(x)\) is odd.

1) Definitions

The decomposition is defined by the universal formulas:

\[ e(x)=\frac{f(x)+f(-x)}{2}, \qquad o(x)=\frac{f(x)-f(-x)}{2}. \]

2) Why this works (quick verification)

Add the two parts: \[ e(x)+o(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}=f(x). \] Also, check symmetry: \[ e(-x)=\frac{f(-x)+f(x)}{2}=e(x)\quad(\text{even}), \qquad o(-x)=\frac{f(-x)-f(x)}{2}=-o(x)\quad(\text{odd}). \]

3) Example

Let \(f(x)=x^2+x\). Then \(f(-x)=x^2-x\). So:

\[ e(x)=\frac{(x^2+x)+(x^2-x)}{2}=x^2, \qquad o(x)=\frac{(x^2+x)-(x^2-x)}{2}=x. \]

The even part is the “symmetric bowl” \(x^2\), and the odd part is the “origin-symmetric line” \(x\).

4) Domain note (important)

The formulas use both \(f(x)\) and \(f(-x)\). If one of them is undefined, then \(e(x)\) and \(o(x)\) may be undefined there. For example, if \(f(x)=\ln(x)\), then \(f(-x)\) is not defined for \(x>0\), so the decomposition is only partially defined.

5) Orthogonality idea (integration prep)

On symmetric intervals \([-a,a]\), even and odd functions behave nicely:

If \(g(x)\) is odd and integrable, then \(\displaystyle \int_{-a}^{a} g(x)\,dx = 0\).
If \(e(x)\) is even, then \(\displaystyle \int_{-a}^{a} e(x)\,dx = 2\int_{0}^{a} e(x)\,dx\).
Products: even·odd is odd, so \(\displaystyle \int_{-a}^{a} \text{(even·odd)}\,dx = 0\).

6) What the calculator shows

The formulas for \(e(x)\) and \(o(x)\) (with your \(f(x)\) substituted).
Graphs of \(e(x)\) and \(o(x)\) (split view), and optionally an overlay with \(f(x)\).
A probe value \(x_0\) showing \(f(x_0)\), \(f(-x_0)\), \(e(x_0)\), \(o(x_0)\), and \(e(x_0)+o(x_0)\).
A recomposition check confirming \(f(x)\approx e(x)+o(x)\) numerically where defined.

Frequently Asked Questions

What is even and odd decomposition of a function?

It writes any function as f(x)=e(x)+o(x), where e(x) is even and o(x) is odd. The parts are defined by e(x)=(f(x)+f(-x))/2 and o(x)=(f(x)-f(-x))/2.

How do I compute the even part e(x) and odd part o(x)?

Compute f(-x) by replacing x with -x in f(x), then form e(x)=(f(x)+f(-x))/2 and o(x)=(f(x)-f(-x))/2. Simplify the expressions if possible to see the structure clearly.

Why can the decomposition be undefined for some x-values?

The formulas require both f(x) and f(-x) to be defined. If either side is not real or not defined at a value, then e(x) and o(x) may also be undefined there.

What does the recomposition check verify?

It tests numerically that e(x)+o(x) matches f(x) within the chosen tolerance across the plotted window where the function is defined. Small differences can occur from sampling and numeric rounding.

How do the graphs help confirm the decomposition?

The even part should be symmetric about the y-axis and the odd part should be symmetric about the origin. Overlaying f(x) and probing x0 and -x0 makes it easy to see how the two parts combine back into f(x).