Mr. Integral
Math notes/Pre-Calculus/Functions Review

Arithmetic on Even and Odd Functions

1 MIN READ · PRACTISE AT YOUR OWN PACE   ·   FREE

LESSON 13 OF 27See the unit map ↗

Predict parity when combining even and odd functions.

The bigger question: What does a function tell us—and what can it hide?

On this page

Idea

Parity rules let you recognize symmetry without expanding a long expression. They require a symmetric common domain; for quotients, remove denominator zeros while retaining that symmetry.

Visual guide

VISUAL GUIDEMultiplication combines symmetry
The even function x² multiplied by the odd function x gives x³, which is odd. Multiplying the odd function x by itself gives x², which is even. Compare reflection and half-turn symmetry.-2-4-1-2001224xy
  • x²: even
  • x³: odd
The even function x² multiplied by the odd function x gives x³, which is odd. Multiplying the odd function x by itself gives x², which is even. Compare reflection and half-turn symmetry.

Method

Even + even is even; odd + odd is odd. Products obey: even × even is even, odd × odd is even, and even × odd is odd. A sum of a nonzero even part and a nonzero odd part is generally neither.

Worked example

The function xsin⁡xx\sin x is even: both xx and sin⁡x\sin x are odd, so their sign changes cancel. The function xcos⁡xx\cos x is odd. For f(x)=x2+xf(x)=x^2+x, its even part is (f(x)+f(−x))/2=x2(f(x)+f(-x))/2=x^2 and its odd part is (f(x)−f(−x))/2=x(f(x)-f(-x))/2=x.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

On a symmetric domain, the product of two odd functions is…

Hint 1 · Find a starting point

Evaluate the product at −x.

Hint 2 · Take the next step

Each odd factor contributes one minus sign.

Show the reasoning

Answer: Even

f(−x)g(−x)=(−f(x))(−g(x))=f(x)g(x).

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

An odd power in a numerator does not determine the parity of an entire rational expression; inspect the denominator too.

Check your understanding

What is the parity of x3/(1+x2)x^3/(1+x^2)?

Show answer

Odd on R\mathbb R: odd numerator divided by an even, nonzero denominator.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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