Odd function

High School

Definition

Defined as an odd function only if f(–x) = –f(x). The graph of an odd function with respect from the origin will be identically symetrical.

Worked examples

\(f(x) = x^3\): \(f(-2) = (-2)^3 = -8\) and \(-f(2) = -(2^3) = -8\), so \(f(-x) = -f(x)\)

Plugging in opposite inputs gives opposite outputs — that's the test for an odd function.

\(f(x) = \frac{1}{x}\): \(f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x)\)

The algebra confirms the odd-function property; the graph is symmetric through the origin.

Common mistakes

\(f(x) = x^2\) is odd because it has symmetry → \(f(x) = x^2\) is even: \(f(-x) = f(x)\), not \(-f(x)\) Symmetry across the y-axis is even; odd functions are symmetric about the origin (180° rotation).
If \(f(-3) = -5\), then \(f(3) = -5\) for an odd function → If \(f(-3) = -5\), then \(f(3) = 5\) For odd functions, \(f(-x) = -f(x)\) means opposite inputs give opposite outputs.
All functions are either even or odd → Most functions are neither even nor odd Only functions satisfying \(f(-x) = f(x)\) or \(f(-x) = -f(x)\) for all x are even/odd.

Where you'll use it next

Odd functions reappear when analyzing polynomial and rational functions, solving integrals over symmetric intervals (where odd functions integrate to zero), and studying Fourier series in calculus and engineering.

Found in 1 StudyPug lesson

Even and Odd Functions: Mastering Identification and Application

Grade 12 Math

Unlock the power of even and odd functions! Learn foolproof methods to identify, analyze, and apply these crucial mathematical concepts. Boost your problem-solving skills and excel in algebra and calculus.

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