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Symmetry Checker

Test if a function is even (f(-x)=f(x)), odd (f(-x)=-f(x)), or neither

Select Function Type

Linear ax Quadratic ax^2+c Cubic ax^3 Sine a*sin(x)

f(x)=ax

f(x)=x

Symmetry Rules

Even: f(-x)=f(x), symmetric about y-axis

Odd: f(-x)=-f(x), origin symmetry

Only even powers (x^2n): even function

Only odd powers (x^(2n+1)): odd function

Function symmetry reveals whether a graph is reflected across the y-axis (even) or rotated 180 about the origin (odd). Understanding symmetry simplifies integration, Fourier analysis, and curve sketching significantly.

⚠ A function can be neither even nor odd - most functions have no symmetry. Only functions with balanced powers exhibit symmetry.

What Is Function Symmetry?

Even functions mirror across the y-axis. Odd functions rotate around the origin. To test: substitute -x for x and simplify. Compare with original f(x) and with -f(x). The result determines the symmetry type.

Even Functions

f(-x)=f(x). Examples: x^2, x^4, cos(x), |x|. Only even powers. Y-axis symmetry.

Odd Functions

f(-x)=-f(x). Examples: x, x^3, sin(x), tan(x). Only odd powers. Origin symmetry.

Test Method

Replace x with -x. If f(-x)=f(x): even. If f(-x)=-f(x): odd. If neither: no symmetry.

Integration Benefits

Even integral from -a to a = 2*int_0^a. Odd integral from -a to a = 0. Major simplification.

Teaching Example: f(x)=x^2. f(-x)=(-x)^2=x^2=f(x) -> EVEN. f(x)=x^3. f(-x)=(-x)^3=-x^3=-f(x) -> ODD. f(x)=x+1. f(-x)=-x+1, neither f(x) nor -f(x) -> NEITHER.

Applications

Integration Fourier Series Graphing Physics Signal Processing

Frequently Asked Questions

Even vs odd?▼

Even: f(-x)=f(x), y-axis symmetry. Odd: f(-x)=-f(x), origin symmetry. Test by substituting -x.

Is x an even or odd function?▼

f(x)=x is odd: f(-x)=-x=-f(x). f(x)=x^2 is even: f(-x)=x^2=f(x). f(x)=x+1 is neither.

Is sin(x) even or odd?▼

sin(x) is odd: sin(-x)=-sin(x). cos(x) is even: cos(-x)=cos(x). tan(x) is odd: tan(-x)=-tan(x).

Integration with symmetry?▼

Even: int_{-a}^{a} = 2*int_{0}^{a}. Odd: int_{-a}^{a} = 0. Huge time-saver in calculus.

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