Even and Odd Functions | Moosa Academy

Even and Odd Functions — Moosa Academy

Numbers are even or odd; so are some functions. The classification rests on symmetry — how a function behaves when the sign of its input is flipped — and it is not merely descriptive. Knowing which kind you have can remove half the work from an integral.

Concept The two conditions

\text{even: } f(-x) = f(x) \qquad \text{odd: } f(-x) = -f(x)

Even — symmetric about the y-axis. Examples:

\cos x

x^2

|x|

Odd — symmetric about the origin. Examples:

\sin x

x^3

x

Most functions are neither. $f(x) = x^2 + x$ mixes an even term with an odd one and so satisfies no condition.

Example Testing a function

Classify $f(x) = x^3$ .

Replace

x

with

-x

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

Compare with

f(x) = x^3

and with

-f(x) = -x^3

It matches the second, so the odd condition holds.

Check numerically:

f(2) = 8

and

f(-2) = -8

⟹ x³ is odd

The method is always the same: substitute $-x$ , simplify, and see which of the two forms you land on.

Example A function that is neither

Classify $f(x) = x^2 + x$ .

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

Is this

f(x) = x^2 + x

? No.

Is it

-f(x) = -x^2 - x

? No.

⟹ neither even nor odd

This is the ordinary case. Even and odd functions are the exceptions, which is precisely why they are worth naming.

Note The payoff in integration

\int_{-a}^{a} f(x)\,dx = 2\int_{0}^{a} f(x)\,dx \quad \text{(even)}, \qquad \int_{-a}^{a} f(x)\,dx = 0 \quad \text{(odd)}

Over an interval symmetric about zero, an even function contributes the same area on both sides — so compute one side and double it.

An odd function contributes a positive area on one side and an equal negative area on the other. They cancel exactly, and the integral is zero without any calculation at all.

Summary

A function is even when f(−x) = f(x), giving symmetry about the y-axis.
It is odd when f(−x) = −f(x), giving symmetry about the origin.
Most functions are neither; test by substituting −x and simplifying.
On a symmetric interval, an even integral doubles one half and an odd integral is zero.