Test if a function is even (f(-x)=f(x)), odd (f(-x)=-f(x)), or neither
Select Function Type
f(x)=ax
f(x)=x
f(x)=ax^2+c
f(x)=x^2+
f(x)=ax^3
f(x)=x^3
f(x)=a*sin(x)
f(x)=sin(x)
Result
Derivation
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.
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