Odd Functions

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Idea

An odd function has symmetry under a half-turn about the origin. Its domain must be symmetric, and the outputs at opposite inputs must also be opposite.

Visual guide

VISUAL GUIDEOpposite inputs give opposite outputs

For the odd function f(x) = x³, rotating the graph half a turn about the origin leaves it unchanged. The marked points (−1, −1) and (1, 1) satisfy f(−1) = −f(1).

Method

f(-x)=-f(x).

If zero belongs to the domain, oddness forces $f(0)=0$ . That necessary condition alone does not prove a function is odd.

Worked example

For $f(x)=x^3-2x$ , $f(-x)=-x^3+2x=-(x^3-2x)$ . The reciprocal $1/x$ is also odd on its symmetric domain excluding zero. In contrast, $x^3+1$ fails because the constant does not change sign.

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.

Which function is odd on ℝ?

f(x)=x²

f(x)=x³+1

f(x)=x³

Hint 1 · Find a starting point

An odd function satisfies f(−x)=−f(x).

Hint 2 · Take the next step

Check whether a constant offset destroys the identity.

Show the reasoning

Answer: f(x)=x³

(−x)³=−x³; the added 1 in x³+1 would prevent oddness.

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

Common mistake

The zero function is both even and odd on any symmetric domain. Most functions are neither.

Check your understanding

Is $f(x)=x/(1+x^2)$ odd?

Show answer

Yes. The numerator changes sign and the denominator does not.

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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